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Addendum to the paper “Numerical solution of second-order equations on plane domains”. (English) Zbl 0769.65077
In a previous paper [ibid. 25, No. 2, 169-191 (1991; Zbl 0717.65082)] a so-called upwind discretization scheme was applied to the problem \(- \Delta u + b\cdot \nabla u + cu = f\) in \(D\); \(u=0\) on \(B\) where \(D\) is a bounded plane domain with polygonal boundary. For the proof of convergence, it was required that the underlying triangulations be uniformly acute, i.e. in addition to the usual assumption on the existence of a uniform lower bound for the interior angles of the triangles, the author needed a uniform upper bound strictly less than \(\pi/2\). An inverse assumption (quasiuniformity condition) was also necessary.
In this note, the author shows that these last two assumptions are not related to the discretization principle itself, but rather with the approximation quality of the coefficients which can be achieved.
MSC:
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
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References:
[1] [1] L. ANGERMANN, Numerical solution of second-order elliptic equations on plane domains, M2AN Modél. Math. Anal. Numér., 25(2) : 169-191, 1991. Zbl0717.65082 MR1097143 · Zbl 0717.65082
[2] P. CIARLET, The finite element method for elliptic problems, North-Holland, Amsterdam-New York-Oxford, 1978. Zbl0383.65058 MR520174 · Zbl 0383.65058
[3] T. IKEDA, Maximum principle in finite element models for convection-diffusion phenomena, North-Holland, Amsterdam-New York-Oxford, 1983. Zbl0508.65049 MR683102 · Zbl 0508.65049
[4] J. WLOKA, Partielle Differentialgleichungen. Sobolevräume und Randwertaufgaben, Teubner-Verlag, Stuttgart, 1982. Zbl0482.35001 MR652934 · Zbl 0482.35001
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