An adaptive strategy

for elliptic problems including

a posteriori controlled boundary approximation

Authors:
W. Dörfler and M. Rumpf

Journal:
Math. Comp. **67** (1998), 1361-1382

MSC (1991):
Primary 65N15, 65N30, 65N50

MathSciNet review:
1489969

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Abstract | References | Similar Articles | Additional Information

Abstract: We derive a posteriori error estimates for the approximation of linear elliptic problems on domains with piecewise smooth boundary. The numerical solution is assumed to be defined on a Finite Element mesh, whose boundary vertices are located on the boundary of the continuous problem. No assumption is made on a geometrically fitting shape.

A posteriori error estimates are given in the energy norm and the -norm, and efficiency of the adaptive algorithm is proved in the case of a saturated boundary approximation. Furthermore, a strategy is presented to compute the effect of the non-discretized part of the domain on the error starting from a coarse mesh. This especially implies that parts of the domain, where the measured error is small, stay non-discretized. The presented algorithm includes a stable path following to supply a sufficient polygonal approximation of the boundary, the reliable computation of the a posteriori estimates and a mesh adaptation based on Delaunay techniques. Numerical examples illustrate that errors outside the initial discretization will be detected.

**[Ad]**Robert A. Adams,*Sobolev spaces*, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. MR**0450957****[Al]**Alt, H. W. (1985): Lineare Funktionalanalysis. Springer, Berlin.**[Bä]**Eberhard Bänsch,*Local mesh refinement in 2 and 3 dimensions*, Impact Comput. Sci. Engrg.**3**(1991), no. 3, 181–191. MR**1141298**, 10.1016/0899-8248(91)90006-G**[BK]**James H. Bramble and J. Thomas King,*A robust finite element method for nonhomogeneous Dirichlet problems in domains with curved boundaries*, Math. Comp.**63**(1994), no. 207, 1–17. MR**1242055**, 10.1090/S0025-5718-1994-1242055-6**[BR]**I. Babuška and W. C. Rheinboldt,*Error estimates for adaptive finite element computations*, SIAM J. Numer. Anal.**15**(1978), no. 4, 736–754. MR**0483395****[BW]**R. E. Bank and A. Weiser,*Some a posteriori error estimators for elliptic partial differential equations*, Math. Comp.**44**(1985), no. 170, 283–301. MR**777265**, 10.1090/S0025-5718-1985-0777265-X**[Ba]**Baker, T. J. (1989): Automatic mesh generation for complex three-dimensional regions using a constrained Delaunay triangulation. Engineering with Computers, 5, 161-175.**[Ci]**Philippe G. Ciarlet,*The finite element method for elliptic problems*, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. Studies in Mathematics and its Applications, Vol. 4. MR**0520174****[Cl]**Ph. Clément,*Approximation by finite element functions using local regularization*, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. \jname RAIRO Analyse Numérique**9**(1975), no. R-2, 77–84 (English, with Loose French summary). MR**0400739****[Dö1]**Willy Dörfler,*A convergent adaptive algorithm for Poisson’s equation*, SIAM J. Numer. Anal.**33**(1996), no. 3, 1106–1124. MR**1393904**, 10.1137/0733054**[Dö2]**W. Dörfler,*A robust adaptive strategy for the nonlinear Poisson equation*, Computing**55**(1995), no. 4, 289–304 (English, with English and German summaries). MR**1370104**, 10.1007/BF02238484**[GH]**P. L. George and F. Hermeline,*Delaunay’s mesh of a convex polyhedron in dimension 𝑑. Application to arbitrary polyhedra*, Internat. J. Numer. Methods Engrg.**33**(1992), no. 5, 975–995. MR**1153607**, 10.1002/nme.1620330507**[HTH]**B. Hamann, H. J. Thornburg, and G. Hong,*Automatic unstructured grid generation based on iterative point insertion*, Computing**55**(1995), no. 2, 135–161 (English, with English and German summaries). MR**1345245**, 10.1007/BF02238098**[Ka]**Jan Kadlec,*The regularity of the solution of the Poisson problem in a domain whose boundary is similar to that of a convex domain*, Czechoslovak Math. J.**14 (89)**(1964), 386–393 (Russian, with English summary). MR**0170088****[LM]**J.-L. Lions and E. Magenes,*Non-homogeneous boundary value problems and applications. Vol. I*, Springer-Verlag, New York-Heidelberg, 1972. Translated from the French by P. Kenneth; Die Grundlehren der mathematischen Wissenschaften, Band 181. MR**0350177****[Re]**Rebay, S. (1993): Efficient unstructured mesh generation by means of Delaunay triangulation and Bowyer-Watson algorithm. J. Comput. Physics, 106, 125-138.**[Ri]**M.-Cecilia Rivara,*Algorithms for refining triangular grids suitable for adaptive and multigrid techniques*, Internat. J. Numer. Methods Engrg.**20**(1984), no. 4, 745–756. MR**739618**, 10.1002/nme.1620200412**[SZ]**L. Ridgway Scott and Shangyou Zhang,*Finite element interpolation of nonsmooth functions satisfying boundary conditions*, Math. Comp.**54**(1990), no. 190, 483–493. MR**1011446**, 10.1090/S0025-5718-1990-1011446-7**[SM]**Jerome Spanier and Earl H. Maize,*Quasi-random methods for estimating integrals using relatively small samples*, SIAM Rev.**36**(1994), no. 1, 18–44. MR**1267048**, 10.1137/1036002**[Ve1]**R. Verfürth,*A posteriori error estimation and adaptive mesh-refinement techniques*, Proceedings of the Fifth International Congress on Computational and Applied Mathematics (Leuven, 1992), 1994, pp. 67–83. MR**1284252**, 10.1016/0377-0427(94)90290-9**[Ve2]**R. Verfürth,*A posteriori error estimates for nonlinear problems. Finite element discretizations of elliptic equations*, Math. Comp.**62**(1994), no. 206, 445–475. MR**1213837**, 10.1090/S0025-5718-1994-1213837-1**[We]**Weatherhill, N. P. (1992): Delaunay triangulation in CFD. Comput. Math. Appl., 24.2, 129-150.**[Y]**Harry Yserentant,*On the multilevel splitting of finite element spaces*, Numer. Math.**49**(1986), no. 4, 379–412. MR**853662**, 10.1007/BF01389538

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Additional Information

**W. Dörfler**

Affiliation:
Institut für Angewandte Mathematik, Universität Freiburg, Hermann-Herder- Strasse 10, D-79104 Freiburg, Germany

Email:
willy@mathematik.uni-freiburg.de

**M. Rumpf**

Affiliation:
Institut für Angewandte Mathematik, Universität Bonn, Wegelerstrasse 6, D-52115 Bonn, Germany

Email:
rumpf@iam.uni-bonn.de

DOI:
http://dx.doi.org/10.1090/S0025-5718-98-00993-4

Keywords:
Adaptive mesh refinement,
a posteriori error estimate,
boundary approximation,
Poisson's equation

Received by editor(s):
March 4, 1996

Received by editor(s) in revised form:
January 23, 1997

Article copyright:
© Copyright 1998
American Mathematical Society