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Foundations of Computational Mathematics

Erschienen in:

01.03.2017

Discrete ABP Estimate and Convergence Rates for Linear Elliptic Equations in Non-divergence Form

verfasst von: Ricardo H. Nochetto, Wujun Zhang

Erschienen in: Foundations of Computational Mathematics | Ausgabe 3/2018

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Abstract

We design a two-scale finite element method (FEM) for linear elliptic PDEs in non-divergence form $A(x) : D^2 u(x) = f(x)$ in a bounded but not necessarily convex domain $\Omega $ and study it in the max norm. The fine scale is given by the meshsize h, whereas the coarse scale $\epsilon $ is dictated by an integro-differential approximation of the PDE. We show that the FEM satisfies the discrete maximum principle for any uniformly positive definite matrix A provided that the mesh is face weakly acute. We establish a discrete Alexandroff–Bakelman–Pucci (ABP) estimate which is suitable for finite element analysis. Its proof relies on a discrete Alexandroff estimate which expresses the min of a convex piecewise linear function in terms of the measure of its sub-differential, and thus of jumps of its gradient. The discrete ABP estimate leads, under suitable regularity assumptions on A and u, to pointwise error estimates of the form

$$\begin{aligned} \Vert \,u - u^{\epsilon }_h\,\Vert _{L^{\infty }(\Omega )} \le \, C(A,u) \, h^{2\alpha /(2 + \alpha )} \big | \ln h \big | \qquad 0< \alpha \le 2, \end{aligned}$$

provided $\epsilon \approx h^{2/(2+\alpha )}$. Such a convergence rate is at best of order $ h \big | \ln h \big |$, which turns out to be quasi-optimal.

Nächster Artikel Exponential Convergence of hp-FEM for Elliptic Problems in Polyhedra: Mixed Boundary Conditions and Anisotropic Polynomial Degrees

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Titel: Discrete ABP Estimate and Convergence Rates for Linear Elliptic Equations in Non-divergence Form
verfasst von: Ricardo H. Nochetto
Wujun Zhang
Publikationsdatum: 01.03.2017
Verlag: Springer US
Erschienen in: Foundations of Computational Mathematics / Ausgabe 3/2018
Print ISSN: 1615-3375
Elektronische ISSN: 1615-3383
DOI: https://doi.org/10.1007/s10208-017-9347-y

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