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

Discrete ABP Estimate and Convergence Rates for Linear Elliptic Equations in Non-divergence Form We design a two-scale finite element method (FEM) for linear elliptic PDEs in non-divergence form $$A(x) : D^2 u(x) = f(x)$$ A ( x ) : D 2 u ( x ) = f ( x ) in a bounded but not necessarily convex domain $$\Omega$$ Ω and study it in the max norm. The fine scale is given by the meshsize h, whereas the coarse scale $$\epsilon$$ ϵ 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} ‖ u - u h ϵ ‖ L ∞ ( Ω ) ≤ C ( A , u ) h 2 α / ( 2 + α ) | ln h | 0 < α ≤ 2 , provided $$\epsilon \approx h^{2/(2+\alpha )}$$ ϵ ≈ h 2 / ( 2 + α ) . Such a convergence rate is at best of order $$h \big | \ln h \big |$$ h | ln h | , which turns out to be quasi-optimal. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Foundations of Computational Mathematics Springer Journals

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

, Volume 18 (3) – Mar 1, 2017
57 pages

/lp/springer_journal/discrete-abp-estimate-and-convergence-rates-for-linear-elliptic-zzqEU8lbak
Publisher
Springer Journals
Subject
Mathematics; Numerical Analysis; Economics, general; Applications of Mathematics; Linear and Multilinear Algebras, Matrix Theory; Math Applications in Computer Science; Computer Science, general
ISSN
1615-3375
eISSN
1615-3383
D.O.I.
10.1007/s10208-017-9347-y
Publisher site
See Article on Publisher Site

### Abstract

We design a two-scale finite element method (FEM) for linear elliptic PDEs in non-divergence form $$A(x) : D^2 u(x) = f(x)$$ A ( x ) : D 2 u ( x ) = f ( x ) in a bounded but not necessarily convex domain $$\Omega$$ Ω and study it in the max norm. The fine scale is given by the meshsize h, whereas the coarse scale $$\epsilon$$ ϵ 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} ‖ u - u h ϵ ‖ L ∞ ( Ω ) ≤ C ( A , u ) h 2 α / ( 2 + α ) | ln h | 0 < α ≤ 2 , provided $$\epsilon \approx h^{2/(2+\alpha )}$$ ϵ ≈ h 2 / ( 2 + α ) . Such a convergence rate is at best of order $$h \big | \ln h \big |$$ h | ln h | , which turns out to be quasi-optimal.

### Journal

Foundations of Computational MathematicsSpringer Journals

Published: Mar 1, 2017

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