On the sharpness of certain local estimates for $\text {\textit {\r {H}}}^1$ projections into finite element spaces: influence of a re-entrant corner
HTML articles powered by AMS MathViewer
- by Lars B. Wahlbin PDF
- Math. Comp. 42 (1984), 1-8 Request permission
Abstract:
In a plane polygonal domain with a reentrant corner, consider a homogeneous Dirichlet problem for Poisson’s equation $- \Delta u = f$ 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, $0 < h \leqslant 1$. Assuming that u has a singularity of the type $|x - {v_M}{|^\beta }$ at the vertex ${v_M}$ of maximal angle $\pi /\beta$, we show: (i) For any subdomain A and any s, the error measured in ${H^{ - s}}(A)$ is not better than $O({h^{2\beta }})$. (ii)On annular strips of points of distance of order d from ${v_M}$, the pointwise error is not better than $O({h^{2\beta }}{d^{ - \beta }})$.References
- Robert A. Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0450957
- Philippe G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR 0520174 M. Dobrowolski, Numerical Approximation of Elliptic Interface and Corner Problems, Rheinischen Friedrich-Wilhelms-Universität, Bonn, 1981. P. Grisvard, Boundary Value Problems in Non-Smooth Domains, Department of Mathematics, University of Maryland, Lecture Notes 19, College Park, MD, 1980.
- R. B. Kellogg, Higher order singularities for interface problems, The mathematical foundations of the finite element method with applications to partial differential equations (Proc. Sympos., Univ. Maryland, Baltimore, Md., 1972) Academic Press, New York, 1972, pp. 589–602. MR 0433926, DOI 10.1007/bf01932971
- V. A. Kondrat′ev, Boundary value problems for elliptic equations in domains with conical or angular points, Trudy Moskov. Mat. Obšč. 16 (1967), 209–292 (Russian). MR 0226187
- Pentti Laasonen, On the discretization error of the Dirichlet problem in a plane region with corners, Ann. Acad. Sci. Fenn. Ser. A I No. 408 (1967), 16. MR 0232009
- Joachim Nitsche, Zur lokalen Konvergenz von Projektionen auf finite Elemente, Approximation theory (Proc. Internat. Colloq., Inst. Angew. Math., Univ. Bonn, Bonn, 1976) Springer, Berlin, 1976, pp. 329–346 (German, with English summary). MR 0658317
- J. A. Nitsche, Der Einfluss von Randsingularitäten beim Ritzschen Verfahren, Numer. Math. 25 (1975/76), no. 3, 263–278. MR 436606, DOI 10.1007/BF01399415
- Joachim A. Nitsche and Alfred H. Schatz, Interior estimates for Ritz-Galerkin methods, Math. Comp. 28 (1974), 937–958. MR 373325, DOI 10.1090/S0025-5718-1974-0373325-9
- A. H. Schatz and L. B. Wahlbin, Maximum norm estimates in the finite element method on plane polygonal domains. I, Math. Comp. 32 (1978), no. 141, 73–109. MR 502065, DOI 10.1090/S0025-5718-1978-0502065-1
- A. H. Schatz and L. B. Wahlbin, On the finite element method for singularly perturbed reaction-diffusion problems in two and one dimensions, Math. Comp. 40 (1983), no. 161, 47–89. MR 679434, DOI 10.1090/S0025-5718-1983-0679434-4
- Robert Schreiber, Finite element methods of high-order accuracy for singular two-point boundary value problems with nonsmooth solutions, SIAM J. Numer. Anal. 17 (1980), no. 4, 547–566. MR 584730, DOI 10.1137/0717047
Additional Information
- © Copyright 1984 American Mathematical Society
- 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