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 and P.-A. Raviart,*Maximum principle and uniform convergence for the finite element method*, Comput. Methods Appl. Mech. Engrg.**2**(1973), 17–31. MR**0375802**, https://doi.org/10.1016/0045-7825(73)90019-4**[2]**Pierre Grisvard,*Behavior of the solutions of an elliptic boundary value problem in a polygonal or polyhedral domain*, Numerical solution of partial differential equations, III (Proc. Third Sympos. (SYNSPADE), Univ. Maryland, College Park, Md., 1975) Academic Press, New York, 1976, pp. 207–274. MR**0466912****[3]**Olga A. Ladyzhenskaya and Nina N. Ural’tseva,*Linear and quasilinear elliptic equations*, Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis, Academic Press, New York-London, 1968. MR**0244627****[4]**H. D. MITTELMAN, "On a discrete maximum principle in the finite element method." (Preprint.)**[5]**Frank Natterer,*Über die punktweise Konvergenz finiter Elemente*, Numer. Math.**25**(1975/76), no. 1, 67–77 (German, with English summary). MR**0474884**, https://doi.org/10.1007/BF01419529**[6]**J. A. Nitsche,*𝐿_{∞}-convergence of finite element approximation*, Journées “Éléments Finis”}, address=Rennes, date=1975, (1975)**[7]**J. A. NITSCHE,*Convergence of Finite Element Approximations*, Proc. Sympos. on Mathematical Aspects of Finite Element Methods, Rome, 1975.**[8]**Joachim A. Nitsche and Alfred H. Schatz,*Interior estimates for Ritz-Galerkin methods*, Math. Comp.**28**(1974), 937–958. MR**0373325**, https://doi.org/10.1090/S0025-5718-1974-0373325-9**[9]**A. H. Schatz and L. B. Wahlbin,*Interior maximum norm estimates for finite element methods*, Math. Comp.**31**(1977), no. 138, 414–442. MR**0431753**, https://doi.org/10.1090/S0025-5718-1977-0431753-X**[10]**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**0502065**, https://doi.org/10.1090/S0025-5718-1978-0502065-1**[11]**A. H. Schatz and L. B. Wahlbin,*Maximum norm estimates in the finite element method on plane polygonal domains. II. Refinements*, Math. Comp.**33**(1979), no. 146, 465–492. MR**0502067**, https://doi.org/10.1090/S0025-5718-1979-0502067-6**[12]**Ridgway Scott,*Optimal 𝐿^{∞} estimates for the finite element method on irregular meshes*, Math. Comp.**30**(1976), no. 136, 681–697. MR**0436617**, https://doi.org/10.1090/S0025-5718-1976-0436617-2**[13]**Gilbert Strang and George J. Fix,*An analysis of the finite element method*, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1973. Prentice-Hall Series in Automatic Computation. MR**0443377**

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

Article copyright:
© Copyright 1980
American Mathematical Society