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

Loading next page...
 
/lp/springer_journal/discrete-abp-estimate-and-convergence-rates-for-linear-elliptic-zzqEU8lbak
Publisher
Springer Journals
Copyright
Copyright © 2017 by SFoCM
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

References

You’re reading a free preview. Subscribe to read the entire article.


DeepDyve is your
personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

Enjoy affordable access to
over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month

Explore the DeepDyve Library

Search

Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly

Organize

Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place.

Access

Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals.

Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

See the journals in your area

DeepDyve

Freelancer

DeepDyve

Pro

Price

FREE

$49/month
$360/year

Save searches from
Google Scholar,
PubMed

Create lists to
organize your research

Export lists, citations

Read DeepDyve articles

Abstract access only

Unlimited access to over
18 million full-text articles

Print

20 pages / month

PDF Discount

20% off