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Johnson1990-AdaptiveFEM-ConvectionDiffusion.pdf
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 82 (1990) 301-322
NORTH-HOLLAND
ADAPTIVE FINITE ELEMENT METHODS FOR DIFFUSION AND
CONVECTION PROBLEMS
Claes J O H N S O N
Mathematics Department, Chalmers University of Technology, 412 96 G6teborg, Sweden
Received 10 January 1~190
We give a survey of recent results obtained together with K. Eriksson on adaptive h-methods for the
basic linear partial differential equations of elliptic, parabolic and hyperbolic type. Our adaptive
algorithms are based on a posteriori error estimates leading to reliable methods, and comparison with
sharp a priori error estimates is made to prove efficiencyof the procedures.
I. Introduction
In this note we give a survey of some recent results on adaptive finite element methods
obtained in collaboration with Eriksson, (see [1-8]). As model problems we shall consider the
heat equation including the corresponding stationary Poisson equation representing diffusiondominated problems, and also linear convection-dominated convection-diffusion problems.
Together, these problems cover the basic linear partial differential equations of parabolic,
elliptic and (first order) hyperbolic type. In each of these cases our goal is to solve the
following problem (Problem A): Given a norm II. II, a tolerance TOL > 0, and a piecewise
polynomial finite element discretization of a certain type (e.g., piecewise polynomials of a
certain given degree), design an algorithm for constructing a mesh T with (nearly) minimal
number of degrees of freedom, such that
Ilu- fll
WOt,
(0.1)
where u is the exact solution and U is the finite element solution on the mesh T. Clearly,
Problem A has two ingredients. First, we want the adaptive algorithm to be reliable in the
sense that the error control (0.1) is guaranteed. Secondly, we want the algorithm to be
efficient in the sense that the constructed mesh is nowhere overly refined. Note that for
definiteness our criterion for efficiency is a minimal number of degrees of freedom. Of course,
in practice depending on the particular implementation, mesh generator, solution techniques,
etc., we may accept a certain over-refinement.
Adaptive codes are now entering into applications and adaptivity may be expected to
become a standard feature of finite element software in future. Quantitative error control is of
obvious interest in applications, and efficient techniques for adaptive local refinement or
orientation of the mesh opens fascinating possibilities of computing accurate solutions to
complex problems involving different scales, such as problems in fluid mechancis with
0045-7825/90/$03.50 © 1990- Elsevier Science Publishers B.V. (North-Holland)
302
C. Johnson, Adaptive finite element methods for diffusion and convection problems
boundary layers and shocks, crack problems in solid mechanics, semiconductor problems,
reaction-diffusion problems, etc.
Our adaptive algorithms are based on a posteriori error estimates of the form
Ilu - uII
h, data),
(0.2)
where as indicated the error bound ~ depends on the computed solution U, on the mesh size h
of the corresponding mesh T and the data of the problem. Here h is a function giving the local
mesh size in space and time. Starting from (0.2) we have the following adaptive method for
error control in the II. [1-norm to the tolerance TOL: Find a mesh T, with mesh function h and
corresponding approximate solution U, with minimal number of degrees of freedom such that
~(U, h, data)~<TOL. Since U depends on h, this is a (complex) non-linear minimization
problem. To solve this minimization problem approximately we design an adaptive algorithm
usually of the following form: Given a first coarse mesh T 0, construct successively meshes Tj,
j = 1 , . . . , J, with corresponding mesh functions hj and approximate solutions Uj, with
minimal number of degrees of freedom, such that
~(Uj_I, hi, data) ~< 0 T O L ,
(o.3)
until ~(Uj, h j, data)~<TOL, which is the stopping criterion. Here 0 is a factor ( 0 - 1 )
influencing the total number of steps J required to reach the stopping criterion. Since Uj_I is
given in (0.3), the minimization problem in hj is easy to solve approximately by seeking to
equidistribute the element contributions to the global quantity ~. Note that we consider here
adaptive forms of the so called h-method, where the quantity determined adaptively is the
local element size. More generally, it is of interest to develop methods where the mesh
orientation and stretching, and the degee of the piecewise polynomials are also determined
adaptively.
Since our adaptive algorithms are based on a posteriori error estimates, it follows that the
algorithms are reliable in the above sense; if the stopping criterion ~(U, h, data)~< TOL is
satisfied, then by (0.2) we will have [lu- u[I ~<TOL and the error will be within the given
tolerance. Concerning the efficiency of the adaptive algorithms more or less precise results
may be stated. Ideally, one would like to prove that the final mesh generated by the algorithm
is close to the optimal mesh, which we take to be the mesh with fewest degrees of freedom
such that the approximate solution is within the given tolerance. In certain cases it is possible
to actually prove such a precise result (up to constants of moderate size), while in other cases
we obtain weaker results. In general, to prove efficiency we rely on sharp a priori error
estimates, where the error IIu - u II is estimated by a quantity E(u, h) depending on the exact
solution and the mesh size h. In the elliptic and parabolic case we prove that the a posteriori
quantity ~(U, h, data) may be estimated by a constant times the a priori quantity E(u, h),
which proves efficiency in a weak sense, and may be sharpened through various localization
results to prove efficiency in a strict sense for certain problems. Note that by proving that the a
posteriori bounds may be estimated by a constant times the a priori bounds, it follows in
particular that by decreasing the mesh size it is possible to realize the stopping criterion under
some assumption on the nature of the exact solution, which is not evident from the start.
Summing up so far, our adaptive algorithms are based on (sharp) a posteriori error
estimates leading to reliable methods, and comparison with sharp a priori error estimates is
C. Johnson, Adaptive finite element methods for diffusion and convection problems
303
used to prove efficiency in a more or less precise way. The error estimates are based on a
representation of the error in terms of the solution of a certain dual problem. This error
representation is fundamental in our approach to adaptivity since it gives information on the
structure of the global error as composed of contributions from individual elements, which
gives the basis for the design of the adaptive algorithm. Basically, the error estimates are
obtained by using the orthogonality properties of the Galerkin method and standard finite
element interpolation estimates, together with appropriate stability estimates for the dual
problem. In the case of the a posteriori error estimates the dual problem is a continuous
problem, while for the a priori estimates the dual problem is discrete. In this framework there
is a close analogy between the a priori and a posterior i error estimates which appears to be
fundamental. In both cases the stability of the dual problem is the critical issue. In general, the
stability of the continuous dual problem connected with the a posteriori estimates is easier to
tackle by analytical tools than that of the discrete dual problem related to the a priori
estimates, and thus in many cases the a posteriori estimate is easier to prove than the a priori
estimate, contrary to a common opionion that a posteriori estimates are more difficult to
obtain. For more general problems (e.g., nonlinear problems or problems with variable
coefficients) the stability of the continuous dual problem cannot be accurately estimated by
analytical means (in particular, one has to determine approximately the size of certain
constants involved) and in these cases one has to build in a computational estimate of the
stability of the dual problem as a part of the adaptive process. For certain problems and
norms, numerical experiments show that this is feasible, while for more general problems
more work is required to obtain reliable and accurate computational estimates of the stability
of the dual problem. We note that this is a fundamental problem which has to be faced,
analytically or computationally, since the stability of the dual problem reflects the error
propagation properties of the given equation.
For parabolic problems we use the Discontinuous Galerkin method based on a space-time
finite element discretization with basis functions continuous in space and discontinuous in
time. The time step and the space discretization may vary from one time level to the next and
it is also possible to use more general space-time meshes with the time steps being variable
also in space. For hyperbolic type problems we use the Streamline Diffusion method
(SD-method) again with space-time finite elements in the time-dependent case.
We now briefly comment on the difference in our approach to adaptivity as compared to the
pioneering work by Babu~ka, see e.g. [9], and the related work by Bank [10] and Ewing [11].
First, in the work by Babu~ka et al., the emphasis is on elliptic problems with error control in
the energy norm, while we consider also other norms and also parabolic and hyperbolic
problems. Secondly, Babu~ka et al. seek to construct a posteriori error bounds which are very
precise in the sense that the quotient between the estimated and the actual global error (the
effectivity index) tends to one as the mesh size tends to zero. For this purpose elaborate a
posteriori estimates based on solving local problems are used. However, in our approach we
set the goal lower in this respect, and we use simpler possibly less precise estimates and accept
(depending on the difficulty of the problem, the chosen norm and the tolerance) effectivity
indices in the range, say, from 1 to 3. In contrast we are able to attack more general problems
and we are not restricted to only energy norms. A further difference is that we seek to obtain
adaptive algorithms which we can prove to be efficient in a more or less precise way. Let us
note that we should distinguish between the concept of effectivity index and the efficiency of
304
C. Johnson, Adaptive finite element methods for diffusion and convection problems
the adaptive algorithm. Even if the effectivity index is close to one, which says that we are
able to estimate the global error on a given mesh very accurately, it is not clear that the
underlying mesh is close to the optimal mesh related to the corresponding tolerance level; the
given mesh may be locally overly refined and there is no way we can detect this by only
looking at the effectivity index.
To sum up, in our approach we do not seek to achieve effectivity indices necessarily very
close to one, but we seek adaptive algorithms for a general class of problems with error
control in a variety of norms and we seek to prove that the algorithms are efficient in the sense
that almost optimal not overly refined meshes are generated. Note that for parabolic problems
our adaptive methods seem to be the first to give reliable and efficient error control in L~(L2),
i.e., the maximum norm in time and L 2 in space. For hyperbolic problems our results appear
to give the first adaptive methods based on a posteriori error estimates.
After [6] was completed we discovered that our a posteriori error estimates in the energy
norm (Hi-norm) for the Poisson equation are analogous to those presented in Abdalass [12]
and Veffiirth [13] for the Stokes problem. We also learned that similar a posteriori error
estimates for the Poisson equation were already considered in 1979 by Bank [14]. These
estimates are based on estimating the H-1-norm of the residual of the finite element solution
in terms of a weighted L2-norm of the residual over the element interiors and the jumps of the
normal derivatives of the finite element solution across interelement boundaries (using the
orthogonality relation built in the Galerkin method). In this approach (which is very simple
and natural) the residual of the finite element solution is separately estimated in the interior of
the elements and on element boundaries, which leads to effectivity indices not necessarily very
close to one. The development in the early and mid eighties with mathematical emphasis,
however, took a different route concentrating on methods with effectivity index close to one.
Our interest in a posteriori estimates of the indicated form is motivated by their simplicity and
the possibility of handling problems of different nature and different norms. For pioneering
work on adaptive methods with emphasis on engineering aspects, see also [15, 16].
An outline of the remainder of this article is as follows. In Section 1 we present the
discretization methods for our model problems of elliptic, parabolic and hyperbolic type. In
Section 2 we present the a priori and a posteriori error estimates, in Section 3 we state the
corresponding adaptive algorithms and in Section 4 we discuss their reliability and efficiency.
In Section 5 we indicate the structure of the proofs of the a posteriori and a priori error
estimates and finally in Section 6 we present the results of some numerical experiments.
1. The discretization methods
For simplicity we shall restrict our considerations to some standard model problems of
elliptic, parabolic and hyperbolic type, namely, to find u such that
u(x)=O o n F ,
-Au(x)=f
inO,
u,-Au=f
in g t x R ÷,
[3.Vu+au-div(eVu)=f
u=0
inO,
(1.1)
onFxR
u=g
+,
onF,
u(.,0)=u 0 ing2,
(1.2)
(1.3)
c. Johnson, Adaptive finite element methods for diffusion and convection problems
ut+/3"Vu+o~u-div(evu)=f
ing2xR +
u=g
onFxR
305
+
u(.,0)=u 0 inO,
(1.4)
respectively. H e r e / 2 is a bounded polygonal domain in R 2 with boundary F, R + = (0, ~), A is
the usual Laplacian, u t = Ou/Ot,/3 = (/31,/32) is a given velocity field, a is a given absorption
coefficient, e t> 0 a given (small) diffusion coefficient, all coefficients possibly depend on x and
t, and the functions f, g and u 0 are given data. We note that when e = 0, then the boundary
condition u = g in (1.3) is imposed only on the inflow part of the boundary F_ = {x E
F[ n(x)./3(x) < 0}, where n(x) is the outward unit normal to F at x E F, and similarly for
(1.4).
For the discretization of these problems with respect to the space variable x = (xl, x2), let X
be the class of all finite element discretizations (h, T, S) defined as follows:
(i) h is a positive function in C I ( ~ ) , such that
IVh(x)l
Vx
(1.5a)
(ii) T = {K} is a triangulation of ~ into triangles K of diameter h K, such that
Clh2 ~ fK dx
VK E T,
(1.5b)
and associated with the function h through
c2h K ~
h(x) <~ h K
Vx E K
VK ~ T,
(1.5c)
where c 1 and c 2 are given positive constants,
(iii) S is the set of all continuous functions on 1) which are linear in x on each K E T and
vanish on &O.
As indicated, in the adaptive process we need to construct for a given mesh function h
satisfying (1.5a) a corresponding mesh T satisfying (1.5b,c). In our implementations we have
for this purpose used two mesh generators: one based on successive subdivision of one triangle
into four similar triangles by joining the midpoints of the sides of the given triangle and
introducing 'transition triangles' divided into two subtriangles connecting zones with different
mesh size, and another 'front generator' which constructs a mesh with given local mesh size by
adding elements at a 'front' coinciding initially with the boundary and sweeping the region, cf.
[15].
The stationary elliptic problem (1.1) may now be approximated in the usual way: Let
(h, T h, Sh) ~ 2 and find U E S h such that
(VU, Vv) = ( f, v)
Yv ~ Sh ,
(1.6)
where ( - , - ) denotes the usual inner product i n [L2(O)] d, d = 1, 2. By introducing the L 2
projection operator Ph : L2(I2)---~ S h defined by (Ph w, V)= (W, V) VV ~ Sh, and the discrete
Laplacian Ah:HI(O)---~Sh defined by (AhW, v ) = - ( V w , Vv) Vv E S h, we may write (1.6)
equivalently as --AhU h = Phf, which has a more obvious resemblance with (1.1).
C. Johnson, Adaptive finite element methods for diffusion and convection problems
306
Let us now turn to the time-dependent parabolic problem (1.2). For a full discretization of
this problem with the Discontinuous Galerkin method we consider partitions 0 = t o < t I <
• .. < t, < .-. of N+ into subintervals I, = (tn_l, tn) of length k, = t, - t,_l, and associate with
each such time interval a space discretization (h,,, T,,, Sn) E ~. For q a nonnegative integer we
define Vq,, = { v l v = zq=0 tJq~j, ~0j~ S,}, and discretize (1.2) as follows: Find U such that for
n = 1, 2 , . . . , UIa×1" E Vqn and
f~ {(Ut,v)+(VU, Vv)}dt-t-([V],,_l,v+ ,)=ft
n
(f,v) dt
VV~Vq,,,
(1.7)
tl
where [w], = w + - w~-, w +(-) = lims__,o+(_) w(t, + s) and U o = u 0. W i t h f = 0, (1.7) is equivalent to the sub-diagonal (q + 1, q)-Pad6 scheme of order of accuracy 2q + 1, see [18].
R E M A R K . Note that in the discretization (1.7) the space and time steps may vary in time
and that the space discretization may be variable also in space, whereas the time steps k, are
kept constant in space. Clearly, optimal mesh design requires the time steps to be variable also
in space. Now, it is easy to extend the method (1.7) to admit time steps which are variable in
space simply by defining
u.. = { v Iv(x, t)= E v ,( O x (x) } ,
i
where {)6} is a basis for S, and the coefficients vi now are piecewise polynomial of degree q in
t, without continuity requirements, on partitions of I. which may vary with i. The discrete
functions may now be discontinuous also inside the 'slab' O x I n . The Discontinuous Galerkin
method again takes the form (1.7) with the difference that the term ([U]n_ ~, V + I ) is replaced
by a sum over all jumps of U in 12 x [t,_l, tn) and further the discontinuities of U are
discarded in the integral involving U t. Adaptive methods for the Discontinuous Galerkin
method in this generality are considered in [7].
Finally, we consider the convection diffusion problems (1.3) and (1.4). For the discretization of these problems we shall as indicated use the SD-method which is a variant of a
standard Galerkin finite element methods obtained by two basic modifications: a 'streamline'
modification of the test functions (in the stationary case) from v to v + 8([3 .Vv + a v ) where
8 - h, and a second modification obtained by adding an artificial viscosity term with viscosity
coefficient proportional to h" (with 3 < a < 2) and the residual of the finite element solution.
The streamline SD-method is the first general finite element method for (first order)
hyperbolic equations which combines good stability with high order accuracy. Convergence
results are available for linear scalar convection-diffusion problems, for the incompressible
Euler and Navier-Stokes equations, for scalar conservation laws in several dimensions, and
also (entropy) consistency results for e.g. the compressible Euler and Navier-Stokes equations
(see [19-22]). With q = 1 the SD-method for (1.3) may be formulated as follows in the case
g = 0 a n d e > 0 : Find U ~ S h , s u c h t h a t
([3 . v u + a U , v + 8([3 .Vv + av)) + (~VU, Vv) = ( L v + 8([3 .Vv + av))
Vv ~ s~,
(1.8)
C. Johnson, Adaptive finite element methods for diffusion and convectionproblems
where
(
6 = C 1 max h - ]-fl-[, 0
307
/31 ,
~ = k ( U ) = m a x ( e , C2h~I/3.VU + a U - f ] ) ,
where the C i and v are positive constants with 3 < v < 2. In the computations we normally
choose v close to 2.
For the time-dependent problem (1.4) the SD-method reads as follows using the notation of
(1.7) again with q = 1 and assuming that g = 0 and e > 0 ; Find U, such that for n =
1, 2, . . . . Ula×l" ~ Vln and
fl {(U +/3 .VU + aU, v + 6(v, +/3 .Vv + av))} dt
(1.9)
n
+ ; . ( ~ ' U , Vv) dt + ([U._,],
V;-1)
= I, (f, v + ¢~(Vt + /3 "VV "4- OlV)) dt
where
Vv E V1, ,
d/ h
U o = Uo ,
(
8 = C 1max h
I( 1,/3)1'0
~=~(U)=max(e,
'
)/, (1,/3)1,
C2h~(IUt+/3.VU+aU-fl+
OX 1 ' OX 2
[U"-I] ) )
kn
in12xI.
'
with the C i and v as above.
2. A priori and a posteriori error estimates
In this section we state a priori and a posteriori error estimates for the discretization
methods (1.6)-(1.9). By II II we denote the L2(12)-norm and DSu = (El,,l=s Io=ul2) x'2. For
the stationary elliptic problem (1.1) we have the following a priori estimate.
T H E O R E M 2.1. Let f C L 2 ( 1 2 ) and let u and U be the solutions of (1.1) and (1.6),
respectively. Then for m = 1 and 2 there exists a constant C depending only on the constants c a
and c 2 in (1.5), such that
I I D 2 - " ( u - U)ll ~
CllhmDzu]l,
where for m = 2 we assume that ix is sufficiently small and 12 is convex.
(2.1)
308
C. Johnson, Adaptive finite element methods for diffusion and convection problems
R E M A R K 2.1. Note the way in which the local mesh size h(x) enters in these error estimates
showing that large second derivatives of u may be compensated for by a (locally) small mesh
size so as to control the quantity tIhmD2ul[, m = 1, 2, bounding the error. This indicates the
possibility of adaptively choosing the mesh size to control the error if D2u may be computationally estimated, an idea which was explored in [4]. In this note, however, we will follow a
related but different adaptive strategy directly based on a posteriori error estimates.
R E M A R K 2.2. Note further that the estimate (2.1) is optimal in the sense that there exists a
constant c such that 'for most u' (e.g., if D~u, lal = 2, is roughly constant on each element),
inf [ID2-
(u-v)ll>
uES h
cllhmD2ull
,
m= 1,2,
which indicates that error control based on (2.1) should be efficient.
The error estimate (2.1) with m = 1 is classical, whereas with m = 2 the estimate in the
present generality can be found in [2]. For quasi-uniform partitions (corresponding to taking h
constant) the case m = 2 is well-known. Let us further remark that (2.1) may also be derived
for a (convex or nonconvex) domain g2 with smooth boundary, in the case m = 2 with the
constant C depending on g2.
To state the a posteriori estimate for the stationary elliptic problem (1.1) underlying the
adaptive algorithm we need some notation. With each side ~"E OK n O of a triangle K ~ T h
we associate a vector n, of length one normal to z and define for v E S h
[ O~V~nV]=lim+(Vv(x,_,0+ s n : ) - V v ( x - s n ~ ) ) ' n + '
xEz,
that is, [Ov/On~] is the j u m p across z in the normal c o m p o n e n t of Vv. We define for v ~ S h the
piecewise constant quantity D2hv by
=
max
.c~aKno
-
-
Ol'l¢
hK
on K E
Th ,
where as indicated only the sides ~- in the interior of O occur, which may be viewed as a
discrete counterpart of [D201.
We may then state the following a posteriori error estimates for the stationary elliptic
problem (1.1) given in [6].
T H E O R E M 2.2. There are constants a m and ~m only depending on the constants c 1 and c2,
such that if f E Lz(g2 ) and u and U are the solutions o f (1.1) and (1.6), respectively, then for
m = 1,2,
[Io2-m(u -
u)ll
amllh fll +
[3~llhmOzhUl[,
where in the case m = 2, we assume that g2 is convex.
(2.2)
C. Johnson, Adaptive finite element methods for diffusion and convection problems
309
R E M A R K 2.4. Note that without further analysis the amount of information in the a
posteriori estimate (2.2) is not obvious. Clearly, if we compute U using (1.6), then we may
bound the error using (2.2) by evaluating the right-hand sides of these estimates. If these
quantities turn out to be sufficiently small, then we may be satisfied and quit. However,
without further analysis it is conceivable that the right-hand sides of (2.2) would always be
large and then these estimates would be useless. In fact, a posteriori error estimates of the
form (2.2) may be derived also for unstable methods and in such cases the right-hand side
quantities could be large regardless of the mesh size. In our case we shall prove that in fact
(2.2) is sharp, and thus may be useful in practice, by comparison with the optimal a priori
estimates (2.1).
R E M A R K 2.5.
a3llh4-mD2fl [.
If f E H2(12),
then the f-terms in (2.2) or (2.3) may be replaced by
Let us next state optimal a priori estimates for the parabolic problem (1.2) given in [6]. For
simplicity we assume that 12 is convex. The estimates may be extended to general domains
with smooth boundary with the constants C depending on 12.
T H E O R E M 2.3. Let u be the solution o f ( 1 . 2 ) and U that of(1.7), suppose tx is small enough
and assume that for each n one of the following two assumptions holds:
S, C S , _ , ,
(2.4a)
f~] <<-y k , ,
(2.4b)
where/~, = maXxe~ h,(x) and 7 is sufficiently small and that for all n, k, <- Ck,+,. Then there
exist constants C only depending on c I and c 2 (if 12 is convex) such that for q = 0, 1 and
N=1,2,...
uII1N CLN m a x , , , ~ u
Ilu -
and f o r q = l
(2.5a)
Eq,(U) ,
andN=l,2,...,
Ilu(tN)- u; ll
CLNm a x l ~ . ~
u
E2.(u),
(2.5b)
where
1(
LN=~
Eq.(U) =
tN
)1/2
ln~-~n+l_
'
min k ,j u t(j) in + Ilhu,OZul[, ,
with
U}1) =
q = 0, 1, 2
j~q+l
U t ,
. (2) ~---
IA t
Utt
,
(3)= Au, and
U t
Ilwll,o
.
=
maxllw(/)ll
,
R E M A R K 2.6. Note that (2.5) states that the Discontinuous Galerkin method (1.7) is of
310
c. Johnson, Adaptive finite element methods for diffusion and convection problems
o r d e r q + 1 globally in time and of order 2q + 1 at the discrete time levels t, for q = 0, 1.
Further, the estimates (2.5a, b) are optimal in the sense that for s o m e positive constant c
inf I l u - oil In
V~Vqn
~-CEqn(U)
q=0,1,2
,
(2.6)
,
if here, in the definition of Eqn(U), we put ul 3~ = u,, and restrict the variation of ul 3) and D~'u
for I 1---2 as in R e m a r k 2.2. N o t e that for the 'super approximation' result (2.5b) it is
relevant to c o m p a r e with approximation in V2,.
R E M A R K 2.7. With quasi-uniform space-meshes with h , , ( x ) ~ ft,, we expect to h a v e / ~ 2 _ k ,
for q = 0 and/~2 < < k , if q = 1, since the Discontinuous Galerkin m e t h o d is of second o r d e r
in space and of o r d e r 2q + 1 in time, q = 0, 1. Thus, in particular for q = 1 the condition (2.4b)
does not appear to be restrictive and in fact allows a considerable variation of hn(X ). In certain
extreme situations, however, e.g. with initial data u 0 highly c o n c e n t r a t e d in space, (2.4b) m a y
impose a restriction on the mesh. It is possible that (2.4b) m a y be w e a k e n e d to a condition of
the form/~2 ~< 7 K , , where K, = t,. - t,_ 1 and S m = S, for m = n, n + 1 , . . . , n*.
We now state a posteriori estimates for the parabolic p r o b l e m (1.2) (see [6]). Again, we
assume that O is convex, but generalizations to s m o o t h non-convex domains are possible (cf.
R e m a r k 2.10).
Theorem 2.4. Let u be the solution o f ( 1 . 2 ) and U that of (1.7), and suppose J2 is convex.
Then for N >I 1, we have for q = 0
Ilu(tN) - U;~ll ~
max
~0.(U)
(2.7a)
max
~2n(U),
(2.7b)
l~n~N
and for q = 1
Ilu(tN)- u ; l l ~
where
l~n~N
~o.(U)=Clllh~.fll,.+C2
2 2 - II+c411[U._l]ll
Ilflldt+C311h.O.U.
fl
n
2
+ CsIIhAU~_l]/k.ll*,
~z.(U)
=
c6ll h2hfl[,, + CTk.z f,. IILII dt
+
. 22
c811hnDnUlli
+ min(C911[U~_l]ll, Clok.llA.e.[u~_dll) + Cl~llhZ.[U._x]/knll * ,
where D ] = D h2n and an asterisk indicates that the corresponding term is present only if
S, ~_ S,_ 1. Further the C i are constants given by
C, = a 2 L ,
C2 = L +2,
C 5 = az(L + e x p ( - 1 ) ) ,
C 8 = 2/32(L + 1 ) ,
173 =/32(L + 2 ) ,
C 6 = a2(L + 2 ) ,
C9 = (74,
C4 = L + 1,
C7 = Y3(3qL + % + 1 ) ,
Clo = 3'2L + 3q,
CH = C5 ,
L = max
LN,
N
C. Johnson, Adaptive finite element methods for diffusion and convection problems
311
where ot2 and [~2 are certain constants depending on c 1 and c 2 related to approximatations by
functions in S h, and the T~ are absolute constants related to one-dimensional approximation by
linear functions (see Section 6).
REMARK
Clllh2.fll,.
~0n by Czkn fi n
2.8. The term
term C z fl n Ilfll dt in
with modified constants (~i.
ClllhaD2fll, ., the
by CTk3n fl,~ IIAft,ll dt
in ~0, and ~2, may be replaced by
Ilf,[I dt and
C7k2n fi n IIf,[I
dt in ~2,
R E M A R K 2.9. The comments of Remark 2.4 are also relevant for the a posteriori estimate
~2.7). By comparison with the optimal a priori estimate (2.5) we can prove that (2.7) is sharp
and thus may be used as a basis for an efficient adaptive algorithm.
R E M A R K 2.10. In the general case with the boundary of O smooth, (some of) the constants
C i should be replaced by constants t~ = CsCi, where C s is a stability constant depending on g2
defined by
Cs =
sup
IID2vll
v~0
The approximation constants a 2 and f12 (depending on c 1 and c2) and the absolute constants Yi
entering in the Ci, may be estimated once and for all (values of these constants used in our
numerical computations are given in Section 9), while the stability constant C s in general
depends on O. It is possible that a relevant value of C s may be found by computing the
quotient IID vll/llAhvll for some properly chosen v ~_ S h. The a posteriori estimates may be
generalized also to problems with variable coefficients or nonlinear problems (see [7]). In this
case the C~ should be replaced by Ci = Cs(u)C~, where Cs(u ) is a 'stability constant' depending
on g2 and the coefficients, and also 'mildly' on u. It is likely that such constants may be
estimated through the adaptive procedure, cf. [1, 5, 7].
R E M A R K 2.11. One can prove direct analogues of Theorems 2.1-2.4, replacing the L2(g'~ )norm by the Lp(g2)-norm, 1 ~ p ~<oo, (see [1]).
Finally, we shall state some a priori and a posteriori error estimates for the SD-method for
the convection-diffusion problems (1.3) and (1.4). We start with an a priori error estimate
from [19] for the stationary problem (1.3), for simplicity with a bounded below by a positive
constant. Further, for notational simplicity we consider the convection dominated case with
e < h. The estimate can easily be extended to a general e to give estimates analogous to (2.2)
in the case e = 1.
T H E O R E M 2.5. Suppose there are positive constants Ki, such that K0 < a(x) < K1, x ~ ~-~, and
suppose the velocity fl is smooth. I f the exact solution u o f (1.3) belongs to W I'=(I2), then there
exists a constant C, such that if e < h, then
118"2( .V(u - u))ll + [[~lnv(u - u)ll + Ilu- uII
where U is the solution o f (1.8).
CIIh =OZull,
C. Johnson, Adaptive finite element methods for diffusion and convection problems
312
We now state an a posteriori error estimate for the SD-method (1.8) for the stationary
problem (1.3) from [7]. For simplicity we shall compare the computed solution U with the
solution a of a perturbed continuous convection-diffusion problem obtained by replacing e by
~(U) in (1.3). It is also possible in model cases to estimate the perturbation error Ilu - all in
terms of k ( U ) - e , U and f, see [8]. In general, we expect 1In- all to be dominated by
C l l a - v i i , so that control of I l a - V I I suffices. In the adaptive algorithm for (1.8) to be
presented below, we also have the option of including the requirement ~ = e, corresponding to
resolution of all details of the exact solution, in which case on the final mesh a = u, see
Section 3. For simplicity we assume that the coefficients a and/3 are constant.
THEOREM
2.6. There is a constant C, such that
Ila- uII ~
C[[min(1, UahZ)R(U)I] + max ~/2,
(2.8)
r_
where
R ( U ) = I/3 . V V + a U -
fl +
IdiVh(~VU)l,
ooomaxmax [,
on
We also state the following analogue of Theorem 2.6 for the SD-method (1.9) for the time
dependent problem (1.4), again assuming for simplicity that a and/3 are constant.
THEOREM
2. 7. There is a constant C, such that
lid - UIIL2(Q> ~ Eli min(1,
Ulh=)R(U)IILz(O)+ mQ_a x
~1/2,
(2.9)
where Q = 0 x I, with I = (0, T) a given time-interval, Q_ = (F_ x I) U (O × {0}),
R(U)=IUt+/3.VU+otU-fI+IdiVh.(~VU)[+ LI
~ II
o n O x I
n .
that (2.8) has essentially (if u = 2 ) the form I l a - u I l ~
should be compared with the a posteriori estimate for the standard
Galerkin method for (1.3) corresponding to choosing 6 = 0 and ~ = e in (1.8): I l u - u I I ~<
CIIR(U)II. In a situation with boundary or internal layers IIR(U)ll~oo as h ~ 0 (until h < e)
REMARK
2.12. Note
CIImin(R(U), 1)11 which
and then adaptive error control is not possible for the standard Galerkin method (unless
h < e), cf. Section 4.
3. Adaptive algorithms
In this section we present the adaptive algorithms based on the a posteriori error estimates
stated above, considering first the elliptic problem (1.1). Starting from the a posteriori error
estimate (2.2) we have the following algorithm for control of 11D2-m(u - U)[I, m = 1, 2: Given
an initial triangulation To, determine successively triangulations Tj with Nj elements and mesh
functions hi and corresponding approximate solutions Uj, ] = 1 . . . . . . J, such that hj is
C. Johnson, Adaptive finite element methods for diffusion and convection problems
313
maximal under the condition
mllh mfllL
., +t mll
hinD 2 H
j
~< 0TOL
K~Tj_I,
(3.1)
where J is the smallest integer such that
m
m
2
(3.2)
amllhJ fll +/3m]lhJ Dh,_,ujll ~<TOL.
Further, 0 is a parameter (here 0 - ½), through which we may monitor the total number of
steps J required to achieve (3.2). Normally, we may expect to have J - 2 - 5 . Notice that (3.1)
seeks to equidistribute the contribution form each element to the global error bound
amllh=fll + flmllO2ull.
For the parabolic problem the a posteriori error estimates (2.7a, b) have the form
IluN-
U~I I ~<max
~(U, h., k., f),
n~N
(3.3)
where ~ is a quantity related to time step n. The adaptive algorithm based on (3.3) for
control of max~Nl[u n - U~ II has the following form: For n = 1, 2 , . . . , N, construct a mesh
S~ with Nn elements and mesh function hn, a time step k, and corresponding approximate
solutions U~ on $2 x I~, such that
~n(Un, hn, kn,
f)
=
TOL,
and Nn/k . is (nearly) minimal. To solve the minimization problem we again seek to
equidistribute the contributions from the elements in the space-time discretization o f / 2 x I,.
For a precise statement of the adaptive algorithm in this case, see Example 6.2.
For the stationary hyperbolic problem (1.3) discretized by the SD-method (1.8), we may
design two adaptive algorithms: (i) one algorithm based on (2.8) and (ii) one algorithm based
on (2.8) together with the additional requirement that the mesh is refined until ~ = e. In the
case (i) the adaptive algorithm is obtained by replacing (3.1) by
Chj min(1, e]_lhj_l)g(Uj_l)
"
2
~ 0TOL
~ ~
Chjv/2R(Uj_I) 1/2 ~<TOL
if K N F ~ O .
on K E Tj_I,
(3.4a)
(3.4b)
In the case (ii) we also add the requirement that
ChjR(Uj_I)<.e.
(3.4c)
The stopping criterions are obvious. With proper normalization it appears that k = e corresponds to resolving all scales of the continuous solution, see Section 4.
Extension to the time-dependent hyperbolic problem is obtained by replacing/2 by O x I
according to (2.9), and also here we may add the requirement k = e.
C. Johnson, Adaptive finite element methods for diffusion and convection problems
314
REMARK 3.I. For error control in the maximum norm (L~(O)-norm), e.g., for the Poisson
equation, the adaptive algorithm has the form (see [2])
2
2
c(llh=jfllL ¢, + IlhjDhj_
TOL
on KE
Tj_ .
4. Reliability and efficiency
We recall that our adaptive algorithms are based on a posteriori error estimates of the form
Ilu-UIl<.~(U,h, data) and that the stopping criterion is ~ ( U , h , data)~<TOL, which
guarantees that if the stopping criterion is satisfied, then the error will be within the given
tolerance and thus the adaptive algorithm is reliable.
Next, we consider the efficiency of our adaptive algorithms. To prove the efficiency in a
precise way we need to prove that the final mesh produced by the adaptive algorithm is close
to the optimal mesh, which is the mesh with fewest degrees of freedom such that the
correponding approximate solution is within the tolerance. It is possible to show that, e.g., for
the Poisson equation with error control in the maximum norm by using localization techniques
to prove that (see [2]) on a mesh produced by the adaptive algorithm, we have for x E O,
max
ly-xl<~Ch
Ih2(y)D2u(y)l~cTOL.
This result proves essentially that for all x E O the local interpolation error is bounded below
by a constant times the tolerance and thus that the mesh is nowhere overly refined. In
L2-norms efficiency in this precise sense is more difficult to prove and in such cases it may be
of interest to prove efficiency in a weaker sense. We present a simple result of this type for the
Poisson equation, stating that the a posteriori error bounds may be estimated by (sharp) a
priori error bounds (see [6]).
THEOREM 4.1. Under the assumption of Theorem 2.1 there is a constant C, such that for
m = 1,2,
2
mllhmfll + mllh m OhUl[
CllhmO2ul[ .
From this result it follows by Remark 2.2 that in general the global L 2 o r HI-interpolation
error on a mesh produced by the adaptive algorithm is not essentially below the given
tolerance, which indicates efficiency in a certain sense (but does not necessarily exclude local
over-refinement). A somewhat different indication on efficiency also follows from Theorem
4.1, namely that an optimal mesh for which CllhmD2ull = TOE, will (up to a constant) be
accepted by the stopping criterion of the adaptive algorithm. In particular it follows that it is
possible to satisfy the stopping criterion for any tolerance, e.g., if [ID2ull is finite.
For the parabolic problem (1.2) one can prove an efficiency result localized in time
corresponding to Theorem 4.1 stating that for almost all time steps n the interpolation error
Eo,(q = 0) or E2,(q = 1) is not essentially below the given tolerance on meshes generated by
the adaptive algorithm, see [6].
C. Johnson, Adaptive finite element methods for diffusion and convection problems
315
Also for the hyperbolic model problems (1.3) and (1.4) certain results indicating efficiency
of our adaptive algorithms are available. Let us give an outline of these results for the
SD-method (1.8) for the stationary problem (1.3). Typically, the exact solution u of (1.3) is
piecewise smooth with a boundary layer of width O(e) at the outflow boundary F÷ = F~F_ and
internal layers of width O(v'~) along streamlines of the velocity field /3, e.g., if the inflow
boundary data is discontinuous. In the typical case the continuous solution u thus has features
on the three different scales O(1), O(x/-g) and O(e) in smooth regions, internal layers and
outflow layers, respectively. Let us now first consider the adaptive algorithm (3.3) for
L2-norm control based on the a posteriori bound (2.8). In this case theoretical and
computational results indicate that the algorithm will produce a mesh with mesh size of order
O(TOL 2) at the outflow boundary, O(TOL 8/3) at an internal layer, and of order O(TOL) in
regions where the exact solution is smooth. This follows using Theorem 2.6 from localization
results showing that the width of the numerical outflow layer is O(h) and the width of the
internal numerical layer is O(h3/4), (see [19, 20]) and the fact that the integrand in (2.8) will
be of order O(1) in the layers, and by Theorem 2.5 of order O(h) in regions where the exact
solution is smooth. Altogether, these results indicate that the algorithm for L2-norm control
will produce a mesh with correct mesh size close to layers and possibly slight over-refinement
in smooth regions, since there the a priori error estimate indictes O(h 3/2) accuracy, while the a
posteriori estimate only gives O(h). Notice in particular that the algorithm is able to handle a
problem with both boundary and interior layers and smooth parts with a balanced attention to
all features. Depending on the tolerance level chosen and e, the algorithm may resolve
internal layers (if TOL ~< O(e3/16)) and also outflow layers (if TOL ~<O(el/2)).
Next, we add an indication to refine if k > e. In an outflow layer we will have ~ = O(h) if
v = 2, and thus ~ = e will require h = O(e) which corresponds to resolution of the outflow
layer of width O(e). In an internal layer, we will have k = O(h 3/2) if v = 2, and thus ~ = e will
require h = O(e2/3), which again corresponds to resolution of the internal layer of width
O(e 1/2) since the width of the numerical layer is 0(h3/4). Of course, the stated results are
qualitative in nature and are only valid up to constants, but indicate that with proper
normalization the requirement ~ = e imposes resolution of all scales of the continuous
solution.
5. The structure of the proofs of the a priori and a posteriori error estimates
In this section we briefly outline the structure of the proofs of our a priori and a posteriori
error estimates. We start from a continuous problem with the following variational formulation: Find u ~ V, such that
B(u, v) = L(v)
Vv
V,
(5.1)
where B(- ,. ) is a continuous bilinear form on V × V, L is a continuous linear form on V and
V is a Hilbert space (e.g., H i ( o ) in the case of (1.1) and (1.3)). Next, we consider a
corresponding discrete problem: Given a finite element space V h C V, find V ~ Vh such that
B(U, v) = L ( v )
Vv E V h .
(5.2)
316
C. Johnson, Adaptive finite element methods for diffusion and convection problems
To prove an a posteriori error estimate in a norm II" II related to the scalar product ( . , - ) , let
~0E V be the solution of the continuous dual problem: Find ~0E V, such that
B(w, ~) = (w, u - U)
Vw e V,
(5.3)
a problem which we assume to be uniquely solvable. Choosing now w = u - U in (5.3) we
have, using (5.1),
Ilu- uII == B(u- U, q~)= B(u, ~ ) - B(U, ~o)= L(~)- B(U, ~) ,
which gives the following error representation formula using (5.2):
Ilu- ull 2= L(~ - ~ ) - B(U, ~ - 5 ) ,
(5.4)
where ~ ~ V h is an interpolant of q~. The idea is now to establish a stability result for the dual
problem (5.3) of the form
IIl~lll~Cllu-Ull,
(5.5)
where the norm II1" III is as strong as possible, and then estimate ¢ - ~ in a weighted norm (as
strong as possible), with weight depending on (a negative power of) the mesh size h, in terms
of cIIl~lll. Inserting this estimate into (5.4) and dividing by Ilu- uII gives an a posteriori
error estimate of the form
Ilu-Ull~(U,h,Z),
where ~g(U, h, L) depends on U, the mesh size h and the data L. Clearly, the stability
estimate (5.5) for the continuous dual problem (5.3) is the critical ingredient; in particular we
have to find, analytically or computationally, a reasonable approximation of the best constant
in (5.5)
The a priori estimate is obtained by introducing the following discrete dual problem: Find
ck E V h, such that
B ( w , 49) = ( w , (J - U ) ,
where 5 E Vh is an interpolant of u. Choosing here w = (J - U ~ V h we get, using (5.1) and
(5.2),
113- uII == B ( 5 - U, ~ ) = B ( 5 - u, ~ ) ,
(5.6)
from which we obtain an estimate for II5 - ull using a (strong) stability estimate for ~b again
of the form (5.5) (but with a different norm II1" III) and standard interpolation error estimates
for 5 - u .
Summing up, the proofs of the a posteriori and a priori error estimates are based on error
representation formulas of the form (5.4) and (5.6) together with strong stability estimates for
C. Johnson, Adaptive finite element methods for diffusion and convection problems
317
the associated continuous and discrete dual problems, and standard interpolation theory is
used to estimate q~ - q~ and u - U, respectively. The right-hand side of (5.4) is clearly related
to the residual of the discrete solution U, while the right-hand side of (5.6) may be viewed as a
truncation error. For the concrete implementation of the above approach, we refer to
[1-8, 23-25].
6. Numerical results
In this section we present the results of some numerical experiments. Here each mesh Tj is
obtained from a previous mesh Tj_I, starting with a given coarse mesh T 0, by either local
refinement dividing certain triangles (fathers) into four similar triangles (sons) by connecting
the midpoints of the sides, or local unrefinement replacing a group of four sons by their
common father. In particular, the minimal mesh size of the mesh Tj is half of that of Tj_~.
E X A M P L E 6.1. We consider the Poisson equation (1.1) on the square ( - 1 , 1) 2 with f = 0
and exact solution u(xl, x2)= arctan(x2/(x 1 + 1)) with nonzero boundary conditions with a
discontinuity at ( - 1 , 0). In Fig. 1 we give the final mesh produced by (3.1) in the case m = 2
choosing a 2 = 0.15,/32 = 0.3 and TOL = 0.01, together with the estimated and actual L2-error
on the successive meshes.
EXAMPLE
6.2. We consider the following adaptive algorithm for the Discontinuous Galer-
LEVEL NODES
1
2
3
4
5
6
7
16
49
169
472
773
924
1080
L2-NORHERROR
APPROX. L2-ERR
0.26991E+80
0.19159E+90
0.O5f174E-Ot
8.47943E-01
8.24135E-81
8.12262E-81
0.61471E-82
8.65388E+00
8.20221E+00
8.18269E+90
9.$1627E-81
0.27886E-81
0.14872E-81
8.99555E-02
Fig. 1. Laplace equation with discontinuous boundary data. Final mesh and level curves of approximate solution.
Actual and estimated L2-error and number of nodes on the sequence of meshes. T O L = 0.01.
C. Johnson, Adaptive finite element methods for diffusion and convection problems
318
Tolerance
Estimated
0.2
\
0.1
%
%
0.0
0
I
5
t
10
I
I5
error
%
True
I
20
error
I~)
25
T I MESTEP NUMBER
T
IM I - 0
P
0.5
B
l
Z
E
O -0
•
0.0
I
I
0.5
1.0
I
1.5
TIME
2.0
2.
'
N
U
H 2000
B
E
R
0
F
E
L
E
H
E
N
TO
S
1000
0
I
5
I
10
I
15
:
20
'~
2;5 ¢
TIMESTEP NUMBER
Time
step
nr.
5.
Fig. 2. Heat equation with approximate delta-function as initial data. Actual and estimated L2-error and number
of elements in space as functions of the timestep, and the size of the timestep as function of time. The space mesh
and level curves and elevation of approximate solution at time step 5. TOL = 0.25.
C. Johnson, Adaptive finite element methods for diffusion and convection problems
319
kin method (1.7) for the parabolic problem (1.2) based on the a posteriori error estimate
(2.7): For n = 1, 2 , . . . , N, given an initial triangulation T,, 0 and an initial time step k,,0,
determine successively triangulations T,,i with N,,j elements and mesh functions h,,~, time
steps k,,~ and corresponding approximate solutions Un, ~ defined on O x I,, j = 1 , . . . , J, such
that with h,,~ and k,,j maximal
c~ maxllh~,,fll,~=~ + c= maxllh2.,,D~.,,_,g.,,_,llL~,o
+ C~
3o,
(c
~
0TOL
= 2x/N,,.j_~, V K E T..~_~ ,
Ilhn'~[f"'J-~]lk"'~-~]lL~(~o
llf,,ll
,., +min(ClollA,,~ ,P.j ,[U,,~ 1]n
,
C911[U",~-l]"-/k3",~-'ll))-
_
TOL
2
,
_
,
I'
C 6 =
q
_
i/klj
,
_
111
,
if q = 1 ,
C411[Un,~-,]n-,/k,,,i-lll)where C~ = 3,
_
TOL
2
if q = 0 ,
0.15, C 7 = 1/36, C 8 = 0.3, C 9 = 2, C~0 = 1/6 and C~ = 0.2. We choose the
-"
I
I ' I 'L~L~trllrrtl'l~
'lj""'.
,
¢
V'V, !/>/2":-~..<;
-:?~~:-~k:
?:~-~!;-.~!:!:
,~ ) , fez;', !-'A>-~" 1: ~- i ; I [ i ] : ) ? ] t .~'~", ..#z I "~
.4.~mF_[.362]:B~_,s:~%q-W3~.~,¢/ : . , / # _ _ , '
level
5
level
6
LEVEL NODES L2-NORM ERROR
1
16
3
4
5
6
142
382
1027
2418
APPROX. L2-ERR
8.29189E+88
8.29654E+88
8. 18379E+88
8. 12156E+08
8. 98524E-81
8. 67763E-81
8. 28838E+88
8.1634IE+98
8,12973E+88
8.18291E+88
i
Fig. 3. Stationary convection-diffusion problem with internal and boundary layer. Adaptivity based on (3.4a,b),
e = 10 -6, T O L = 0 . 1 . Estimated and actual L2-error and number of nodes on sequence of meshes (level 1
corresponds to the initial mesh). Elevation of final solution. Display of mesh and level curves of approximate
solution for indicated levels.
C. Johnson, Adaptive finite element methods for diffusion and convection problems
320
initial data u o to be an 'approximate deltafunction' at x = 0 : u o =250exp(-Ix]:/250) and
12 = (0, 1) 2. We give in Fig. 2 the sequence of time steps, the number of elements in the
triangulation on each time interval, and the L2(12)-error Ilu(t.)- uZII, n = 1, 2 , . . . , in the
case q = 1, together with the space mesh at time step 5. We notice that the actual error is
approximately constant in time and slightly below the given tolerance.
E X A M P L E 6.3. We now give some results for the adaptive algorithms for the streamline
diffusion methods (1.8) for the stationary hyperbolic problem (1.3) based on (i) (3.4a,b) and
(ii) (3.4a,b) together with the additional refinement criterion ~ = e corresponding to (3.4c).
We consider a problem with both internal layer and outflow layer, and with O = (0, 1) 2, f = 0,
/ 3 = ( 2 , 1 ) , o~=0, u(0, x 2 ) = l for 0 < x 2<~1, u(x 1,1)=1 for 0~<x 1 < 1 and u(xl,x~)=O if
x~ = 1 or x~ = 0. In Figs. 3-6 we give some results with the algorithms (i) and (ii) and varying
e and TOL. The constants C~ in (1.8) were chosen as follows: C 1 = 0.5, C 2 = 0.7. Note that the
width of the layer refinement of mesh Tj is related to the width of the numerical layer of the
approximate solution Us_ ~ on mesh Ts_ ~. This is the reason why the width of the refinement of
Ts appears to be too large as compared to the width of the numerical layer of the solution Us.
Note also that the L2-error is computed, for simplicity, by comparison with the exact solution
corresponding to e = 0, which means that the stated L2-error is not precise for refined meshes
and relatively large e.
I
I/
''
r
/i~i\ t
/
/q-F~;T.~
t,/,'Y \1//~
................... ~
level
k ~ ,'~c~
~d/
j
level
5
6
LEVEL NODES L2-NORM ERROR
j
j
1
2
3
4
5
6
25
78
235
639
1383
1973
Fig. 4. A s in Fig. 3 with n o w e = 10 -4, T O L = 0.05.
9.25618E+89
8.28851E+98
9.15816E+89
9.11518E+90
B.93323E-81
8.82784E-81
APPROX. L2-ERR,
8.26499E+89
8.22927E+89
9.18838E+g9
9,11683E+88
9.57850E-81
9.45399E-81
C. Johnson, Adaptive finite element methods for diffusion and convection problems
/
,
tt -
321
-
,
, '1/1//1//1/,¢../X.,t-Pr>~;;FX4~I~,I ~ q
JtJ
,
level
r~-
J
~ S
;-~"
4
level
5
level
-.,.~._-~-"_~,~. ~~
•
/
,,~t/" /
/
/
6
~
,
, , '"i12/',~,'~, ~ i ,I'~,,.' ,,
Ipj>,,,,,,ii~,~%l,,~,:I,.{.i..i~l~W,.l,
h!.4x ~ , C I
?',?~Y
~Y-'I,,"I\I/1/1,,
LE~L NODES L2-NO~ ERROR APPROX.L2-ERR.
j
~
~
-~i
i
f
~
~'~
level
7
level
1
2
3
4
5
16 B.30811£+BB
46 B.219~E+88
151 8.IB960E+08
3~ 0.12859E+08
835 8.18133£+0B
6
982 8.91251E-81
714780.98%%2E-81
8
2135 0.82937E-01
B.29531E+BB
8.23~3E+BB
8.28248E+80
B.14998E+SB
O.73539E-St
B.IB336E+BB
8.62991E-81
8.64859E-01
8
Fig. 5. A s i n Fig. 3 with n o w the adapti~ty based on (3.4a-~,i.e., with the extra requirement ~ = e;e = I0 -3,
T O L = 0.15.
322
C. Johnson, Adaptive finite element methods for diffusion and convection problems
References
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Fly UP