A weak discrete maximum principle and stability of the finite element method in on plane polygonal domains. I

Author:
Alfred H. Schatz

Journal:
Math. Comp. **34** (1980), 77-91

MSC:
Primary 65N30

DOI:
https://doi.org/10.1090/S0025-5718-1980-0551291-3

MathSciNet review:
551291

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Abstract: Let be a polygonal domain in the plane and denote the finite element space of continuous piecewise polynomials of degree defined on a quasi-uniform triangulation of (with triangles roughly of size *h*). It is shown that if is a "discrete harmonic function" then an a priori estimate (a weak maximum principle) of the form

Now let *u* be a continuous function on and be the usual finite element projection of *u* into (with interpolating *u* at the boundary nodes). It is shown that for any

**[1]**P. G. CIARLET & P. A. RAVIART, "Maximum principle and uniform convergence for the finite element method,"*Comput. Methods Appl. Mech. Engrg.*, v. 2, 1973, pp. 17-31. MR**0375802 (51:11992)****[2]**P. GRISVARD, "Behavior of solutions of an elliptic boundary value problem in a polygonal or polyhedral domain," in*Numerical Solution of Partial Differential Equations*III (SYNSPADE 1975), B. Hubbard, Editor, Academic Press, New York, 1975, pp. 207-274. MR**0466912 (57:6786)****[3]**O. A. LADYŽENSKAJA & N. N. URAL'CEVA,*Linear and Quasilinear Elliptic Equations*, Academic Press, New York, 1968. MR**0244627 (39:5941)****[4]**H. D. MITTELMAN, "On a discrete maximum principle in the finite element method." (Preprint.)**[5]**F. NATTERER, "Über die punktweise Konvergenz finiter elemente,"*Numer. Math.*, v. 25, 66-77. MR**0474884 (57:14514)****[6]**J. A. NITSCHE,*Convergence of Finite Element Approximation*, Second Conference on Finite Elements, Rennes, 1975. MR**568857 (81e:65058)****[7]**J. A. NITSCHE,*Convergence of Finite Element Approximations*, Proc. Sympos. on Mathematical Aspects of Finite Element Methods, Rome, 1975.**[8]**J. A. NITSCHE & A. H. SCHATZ, "Interior estimates for Ritz-Galerkin methods,"*Math. Comp.*, v. 28, 1974, pp. 937-958. MR**0373325 (51:9525)****[9]**A. H. SCHATZ & L. B. WAHLBIN, "Interior maximum norm estimates for finite element methods,"*Math. Comp.*, v. 31, 1976, pp. 414-442. MR**0431753 (55:4748)****[10]**A. H. SCHATZ & L. B. WAHLBIN, "Maximum norm estimates in the finite element method on plane polygonal domains. Part I,"*Math. Comp.*, v. 32, 1978, pp. 73-109. MR**0502065 (58:19233a)****[11]**A. H. SCHATZ & L. B. WAHLBIN, "Maximum norm estimates in the finite element method on plane polygonal domains. Part II, Refinements,"*Math. Comp.*, v. 33, 1979, pp. 465-492. MR**0502067 (58:19233b)****[12]**R. SCOTT, "Optimal estimates for the finite element method on irregular meshes,"*Math. Comp.*, v. 30, 1976, pp. 681-697. MR**0436617 (55:9560)****[13]**G. STRANG & G. J. FIX,*An Analysis of the Finite Element Method*, Prentice-Hall, Englewood Cliffs, N. J., 1973. MR**0443377 (56:1747)**

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DOI:
https://doi.org/10.1090/S0025-5718-1980-0551291-3

Article copyright:
© Copyright 1980
American Mathematical Society