hp-DGFEM for Parabolic Evolution Problems with
Applications to Diusion and Viscous Incompressible
Fluid Flow
Dominik Schotzau
Seminar fur Angewandte Mathematik
ETH Zurich, CH-8092 Zurich, Switzerland
schoetz@sam.math.ethz.ch
CALCOLO 37, pp. 59-64, 2000.
The present article summarizes the main results of the author's doctoral dissertation [5] which
addresses two topics in the area of hp Finite Element Methods (FEM) for problems in solid and
uid mechanics (for a survey on hp-FEM see the recent monograph [9]). The thesis consists
of two parts.
Part I: hp-DGFEM for Parabolic Problems
The Discontinuous Galerkin Finite Element Method (DGFEM) is a time discretization technique for parabolic evolution problems of the form
u0(t) + Lu(t) = g(t); t 2 J = (0; T ); u(0) = u0:
(1)
The known error analyses, however, are mainly concerned with the \h-version" DGFEM where
convergence is achieved by rening the time steps at a xed (usually low) approximation
order (see [10] and the references there). In the rst part of the thesis the error analysis of
the DGFEM is extended to the \p-" and \hp-version" context: The p-version approach is
to achieve convergence by increasing the temporal approximation order on xed time steps,
whereas the hp-version combines judiciously h- and p-renement techniques. It is shown that
the hp-DGFEM leads to spectral and even exponential rates of convergence.
The functional analytic framework for (1) is the usual evolution triple X ,! H = H ,! X of
two separable Hilbert spaces X and H with dense and continuous injections. L : X ! X is
a linear \elliptic spatial" operator and u0 2 H . This setting includes convection-diusion and
viscous incompressible uid ow problems. To discretize (1) in time, the interval J = (0; T )
is partitioned into subintervals M = fIm = (tm 1 ; tm)gMm=1 and to each Im there is assigned a
temporal approximation order rm 2 N 0 . These orders are collected in the vector r = frmgMm=1 .
The hp-DGFEM provides a temporal approximation of (1) in the semidiscrete space
V r (M; X ) = fu : J ! X : ujIm 2 P rm (Im; X )g;
1
P rm (Im ; X ) denoting the space of polynomials of degree rm with coecients in X . We have
the following error estimate:
Theorem 1 Let u be the exact solution of (1) and U the semidiscrete DGFEM approximation
in V r (M; X ). Then:
ku
U k2L2 (J ;X )
M
X
C ( km )2(sm +1) maxf1; r
m=1
2
mg
2
(rm + 1 sm) kuk2
(rm + 1 + sm) H sm+1(Im ;X )
for any 0 sm minfrm; s0;m g, s0;m 2 N 0 . Here, km = length(Im ) and the constant C > 0 is
independent of M , sm, km and rm .
The bounds in Theorem 1 are completely explicit in the step sizes km and the approximation
orders rm . They generalize the best previously known error estimates as e.g. in [10]: They
show that the hp-DGFEM gives spectral accuracy in transient problems with smooth time
dependence. The usual h-version convergence rates can be recovered if rm = r is kept xed.
However, the regularity assumption in Theorem 1 is unrealistic in the presence of singularities
induced by non-smooth initial data (or by discontinuities in the right hand side g). To analyze
the structure of start-up singularities at t = 0, the regularity of u0 is described in terms of
intermediate spaces H = (H; X );2, 0 1, dened by the K -method of interpolation. By
the use of Fourier series techniques (if L is selfadjoint) or of classical semigroup theory (if L
is non-selfadjoint) the subsequent analyticity properties of the exact solution are obtained:
Theorem 2 Let L be an \elliptic operator", let the right hand side g in (1) be analytic and
let u0 2 H for some 0 1. Then the solution u of (1) satises
ku(l) (t)k2X Cd2l (2l + 2)t
(2l+1)+ :
Results of this type have been known, see [1]. However, the (basically sharp) explicit dependence of the estimates on l, t and is new and crucial in the hp-DGFEM. There, singular
behaviour as in Theorem 2 is resolved by geometric time partitions and linearly increasing
approximation orders where t0 = 0, tm = M m and rm = bmc for a geometric grading factor
2 (0; 1) and a slope > 0.
Based on Theorem 1 and 2, exponential rates of convergence can be proved for the hp-DGFEM:
Theorem 3 Let L be an \elliptic operator", let the right hand side g in (1) be analytic and
let u0 2 H for some 0 < 1. Let U be the semidiscrete DG approximation of (1) obtained
with a linearly increasing approximation degree vector r on geometrically rened time steps.
Then we get exponential rates of convergence in the sense that
ku U kL2 (J ;X ) C exp( bN 12 )
with constants C and b independent
Pm=1(rm + of1).N , the number of degrees of freedom used for the time
discretization, i.e. N = M
2
In Theorem 3, N is the number of spatial problems that have still to be solved numerically and
can be viewed as a crude measure for the cost of a given time stepping scheme. We remark
that in Theorem 3 no compatibility conditions are imposed on the initial value u0. Theorem 3
is the rst result of this kind with exponential rates of convergence under realistic regularity
assumptions.
In h-version approaches singularities in time can be resolved on algebraically graded time
meshes at (optimal) algebraic convergence rates:
Theorem 4 Let L be an \elliptic operator", let the right hand side g in (1) be analytic and
let u0 2 H for some 0 < 1. Then for every (uniform) approximation order r there exists
a graded time partition M with N elements such that the DG approximation U 2 V r (M; X )
satises ku U kL2 (J :X ) CN (r+1) .
Again, Theorem 4 generalizes typical h-version results under realistic assumptions on the data.
The hp-DGFEM amounts in each time step Im to the solution of an elliptic spatial system of
rm + 1 equations. This is of course very costly, especially for large rm or in three space dimensions. It is therefore of signicant practical relevance that these equations can be decoupled
into rm + 1 independent (complex) reaction-diusion equations of the form
"2Lu + u = f;
with Ckm=rm2 j"j2 Ckm:
(2)
Problem (2) is singularly perturbed for large rm or small km . This perturbation character
requires in general a careful spatial discretization due to boundary layers that have to be
resolved. In polygonal domains there arise in addition corner singularities. However, on
anisotropically and geometrically rened meshes hp-FEM for the spatial problems (2) are
known to be able to approximate such solution components at (robust) exponential rates of
convergence [4, 9]. Consequently:
Theorem 5 If the hp Finite Elements Methods for the solution of the spatial problems in (2)
are based on hp-version mesh-design principles, exponential rates of convergence in time and
space can be achieved.
Finally, numerical results for the heat equation are presented which conrm the theoretical
results of the rst part of the thesis.
Part II: Stable and Stabilized hp-FEM for Stokes Flow
If the DGFEM time-stepping is applied in incompressible uid ow, stationary Navier-Stokes
equations or linearizations thereof have to be solved in each time step Im . These problems are
still of the form (2) and, again, their solutions can exhibit corner singularity or boundary layer
components. The latter are induced by small DGFEM time steps or, more importantly, by
large Reynolds numbers Re. However, since discretizations of the spatial operators are constrained with incompressibility conditions, the appearance of anisotropic or geometric meshes
3
A3
A4
amesh.eps
68 49 mm
A2
geopolygon.eps
59 41 mm
A1
A5 = A0
Figure 1: Left: Geometric boundary layer meshes near convex and reentrant corners. Right:
Local geometric renement towards the vertices.
in hp-methods raises serious stability questions. In the second part of the thesis such stability
issues are analyzed for the linearized (and stationary) Stokes equations in polygonal domains
R 2 that read as follows: Find a velocity eld ~u and a pressure p such that
~u + rp = f~ in ; r ~u = 0 in ; ~u = ~0 on @ :
(3)
It is proved that the S k S k 2 element family in mixed hp-FEM for (3) satises the BabuskaBrezzi stability condition [2] on meshes that allow for the desired renement properties,
anisotropically towards boundaries and geometrically towards corners. The meshes may contain geometric renement with hanging nodes and the approximation degree k can be variable
throughout the mesh. The inf-sup stability constant is shown to be independent of the element
aspect ratio of the anisotropic elements. This result is formulated for the class of \geometric
boundary layer meshes" which are essentially shape regular meshes that are rened in local
patches geometrically or anisotropically. Examples of such meshes are shown on the left-hand
side of Figure 1. The local patches are indicated by bold lines.
Theorem 6 The S k S k 2 elements are divergence stable on \geometric boundary layer me-
shes" with stability constants which are independent of the aspect ratio of the anisotropic
elements.
Theorem 6 - published in [6, 8] - is the rst divergence stability result for high order elements
on anisotropic meshes. In the thesis numerical estimates of inf-sup constants are presented
that indicate the sharpness of Theorem 6.
From an implementational point of view the most convenient choice of the FE spaces in mixed
hp-FEM would be elements of equal polynomial order in the velocity and the pressure, that
is S k S k elements. However, in that case the Babuska-Brezzi condition is not satised. A
possible way to circumvent this condition is to use stabilized Galerkin Least Squares (GLS)
methods as proposed e.g. in [3].
4
In the thesis, the GLS approach is extended to hp-FEM for Stokes ow in polygonal domains
exhibiting corner singularities. This singular solution behavior is resolved on shape regular
meshes that are geometrically rened towards the vertices fAi g of the domain. On the righthand side of Figure 1 such a local geometric renement is illustrated. The remainder of away from the corners is subdivided with a xed quasi-uniform and regular partition. It is
shown that the use of approximation orders that increase linearly with the distance to the
vertices leads to exponential convergence in the hp-GLSFEM:
Theorem 7 Let (~u; p) be the exact solution of the Stokes problem (3) exhibiting corner singularities in a polygonal domain R 2 . Let (~uN ; pN ) be the discrete solution of the GLSFEM
with equal order \S k S k " elements obtained on geometrically rened meshes with linearly
increasing approximation orders. Then exponential rates of convergence are obtained, i.e.
k~u ~uN kH 1 (
) + kp pN kL2(
) C exp( bN 13 ):
Here, N is the total number of degrees of freedom used for the discretization and the constants
C and b are independent of N .
Theorem 7 holds true verbatim for mixed hp-FEM with stable S k S k 2 elements. These
exponential convergence results are conrmed by a series of numerical studies on a L-shaped
domain where the Stokes solutions exhibit corner singularities.
References
[1] H. Amann: Linear and Quasilinear Parabolic Problems, Volume I: Abstract Linear
Theory, Monographs in Mathematics 89, Birkhauser Verlag, Basel, 1995.
[2] F. Brezzi and M. Fortin: Mixed and Hybrid Finite Element Methods, Springer Series in
Computational Mathematics 15, Springer Verlag, New York, 1991.
[3] E. Boillat and R. Stenberg: An hp error analysis of some Galerkin Least Squares Methods for
the elasticity equations, Research Report 21-94, Helsinki University of Technology, 1994.
[4] J.M. Melenk and C. Schwab: hp-FEM for reaction-diusion equations, I: Robust exponential
convergence, SIAM J. Numer. Anal. 35 (4) (1998), 1520-1557.
[5] D. Schotzau: hp-DGFEM for parabolic evolution problems - applications to diusion and viscous
incompressible uid ow, ETH Dissertation No. 13041, ETH Zurich, 1999.
[6] D. Schotzau and C. Schwab: Mixed hp-FEM on anisotropic meshes, Math. Models and Methods
in Applied Sciences 8 (5) (1998), 787-820.
[7] D. Schotzau and C. Schwab: Time discretization of parabolic problems by the hp-version of the
Discontinuous Galerkin Finite Element Method, submitted to SIAM J. Num. Anal., 1999.
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[8] D. Schotzau, C. Schwab and R. Stenberg: Mixed hp-FEM on anisotropic meshes, II: Hanging
nodes and tensor products of boundary layer meshes, accepted for publication in Numerische
Mathematik.
[9] C. Schwab: p- and hp-Finite Element Methods, Oxford University Press, New York, 1998.
[10] V. Thomee: Galerkin Finite Element Methods for Parabolic Problems, Springer Verlag,
New York, 1997.
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