On the sharpness of certain local estimates for *H* projections into finite element spaces: influence of a re-entrant corner

Author:
Lars B. Wahlbin

Journal:
Math. Comp. **42** (1984), 1-8

MSC:
Primary 65N30; Secondary 65N15

DOI:
https://doi.org/10.1090/S0025-5718-1984-0725981-7

MathSciNet review:
725981

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Abstract: In a plane polygonal domain with a reentrant corner, consider a homogeneous Dirichlet problem for Poisson's equation with *f* smooth and the corresponding Galerkin finite element solutions in a family of piecewise polynomial spaces based on quasi-uniform (uniformly regular) triangulations with the diameter of each element comparable to *h*, . Assuming that *u* has a singularity of the type at the vertex of maximal angle , we show: (i) For any subdomain *A* and any *s*, the error measured in is not better than . (ii)On annular strips of points of distance of order *d* from , the pointwise error is not better than .

**[1]**R. A. Adams,*Sobolev Spaces*, Academic Press, New York, 1975. MR**0450957 (56:9247)****[2]**Ph. G. Ciarlet,*The Finite Element Method for Elliptic Problems*, North Holland, Amsterdam, 1978. MR**0520174 (58:25001)****[3]**M. Dobrowolski,*Numerical Approximation of Elliptic Interface and Corner Problems*, Rheinischen Friedrich-Wilhelms-Universität, Bonn, 1981.**[4]**P. Grisvard,*Boundary Value Problems in Non-Smooth Domains*, Department of Mathematics, University of Maryland, Lecture Notes 19, College Park, MD, 1980.**[5]**R. B. Kellogg, "Higher order singularities for interface problems,"*The Mathematical Foundations of the Finite Element Method*(A. K. Aziz, Ed.), Academic Press, New York, 1972, pp. 589-602. MR**0433926 (55:6896)****[6]**V. A. Kondrat'ev, "Boundary value problems for elliptic equations in domains with conical or angular points,"*Trans. Moscow Math. Soc.*, v. 16, 1967, pp. 227-313. MR**0226187 (37:1777)****[7]**P. Laasonen, "On the discretization error of the Dirichlet problem in a plane region with corners,"*Ann. Acad. Sci. Fenn. Ser. A, I Math.*, v. 408, 1967, pp. 1-16. MR**0232009 (38:335)****[8]**J. A. Nitsche, "Zur lokalen Konvergenz von Projektionen auf finite Elemente,"*Approximation Theory*, Lecture Notes in Math., Vol. 556, Springer, Bonn, 1976, pp. 329-346. MR**0658317 (58:31924)****[9]**J. A. Nitsche, "Der Einfluss von Randsingularitäten beim Ritzschen Verfahren,"*Numer. Math.*, v. 25, 1976, pp. 263-278. MR**0436606 (55:9549)****[10]**J. A. Nitsche & A. H. Schatz, "Interior estimates for Ritz-Galerkin methods,"*Math. Comp.*, v. 28, 1974, pp. 937-958. MR**0373325 (51:9525)****[11]**A. H. Schatz & L. B. Wahlbin, "Maximum norm estimates for the finite element method on plane polygonal domains, Part 1,"*Math. Comp.*, v. 32, 1978, pp. 73-109. MR**0502065 (58:19233a)****[12]**A. H. Schatz & L. B. Wahlbin, "On the finite element method for singularity perturbed reaction diffusion problems in two and one dimensions,"*Math. Comp.*, v. 40, 1983, pp. 47-89. MR**679434 (84c:65137)****[13]**R. Schreiber, "Finite element methods of high order accuracy for singular two-point boundary value problems with non-smooth solutions,"*SIAM J. Numer. Anal.*, v. 17, 1980, pp. 547-566. MR**584730 (82b:65139)**

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DOI:
https://doi.org/10.1090/S0025-5718-1984-0725981-7

Article copyright:
© Copyright 1984
American Mathematical Society