%0 Report %1 nochetto2014discrete %A Nochetto, Ricardo H. %A Zhang, Wujun %D 2014 %K %T Discrete ABP Estimate and Convergence Rates for Linear Elliptic Equations in Non-divergence Form %U http://arxiv.org/abs/1411.6036 %X We design a two-scale finite element method (FEM) for linear elliptic PDEs in non-divergence form $A(x) : D^2 u(x) = f(x)$. 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 (DMP) for any uniformly positive definite matrix $A(x)$ provided that the mesh is weakly acute. Combining the DMP with weak operator consistency of the FEM, we establish convergence of the numerical solution $u_h^\epsilon$ to the viscosity solution $u$ as $\epsilon, h\to0$ and $\epsilonC h|h|$. We develop a discrete Alexandroff-Bakelman-Pucci (ABP) estimate which is suitable for finite element analysis. Its proof relies on a geometric interpretation of Alexandroff estimate and control of the measure of the sub-differential of piecewise linear functions in terms of jumps, and thus of the discrete PDE. The discrete ABP estimate leads, under suitable regularity assumptions on $A$ and $u$, to the estimate \ \| u - u^\epsilon_h \|_L^ınfty(Ømega) \; C \big( h^2 |h| )^/(2 + \alpha) 0< 2, \ provided $(h^2 |łn h|)^1/(2+\alpha)$. Such a convergence rate is at best of order $ h |łn h|^1/2$, which turns out to be quasi-optimal.

@techreport{nochetto2014discrete, abstract = {We design a two-scale finite element method (FEM) for linear elliptic PDEs in non-divergence form $A(x) : D^2 u(x) = f(x)$. 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 (DMP) for any uniformly positive definite matrix $A(x)$ provided that the mesh is weakly acute. Combining the DMP with weak operator consistency of the FEM, we establish convergence of the numerical solution $u_h^\epsilon$ to the viscosity solution $u$ as $\epsilon, h\to0$ and $\epsilon\ge C h|\ln h|$. We develop a discrete Alexandroff-Bakelman-Pucci (ABP) estimate which is suitable for finite element analysis. Its proof relies on a geometric interpretation of Alexandroff estimate and control of the measure of the sub-differential of piecewise linear functions in terms of jumps, and thus of the discrete PDE. The discrete ABP estimate leads, under suitable regularity assumptions on $A$ and $u$, to the estimate \[ \| u - u^{\epsilon}_h \|_{L^{\infty}(\Omega)} \leq \; C \big( h^2 |\ln h| \big )^{\alpha /(2 + \alpha)} \qquad 0< \alpha \leq 2, \] provided $\epsilon \approx (h^2 |\ln h|)^{1/(2+\alpha)}$. Such a convergence rate is at best of order $ h |\ln h|^{1/2}$, which turns out to be quasi-optimal.}, added-at = {2014-12-04T01:40:42.000+0100}, author = {Nochetto, Ricardo H. and Zhang, Wujun}, biburl = {https://www.bibsonomy.org/bibtex/24f0afd48875c631c876c977f03da9cc1/cerniagigante}, interhash = {204be4e250022637b6b5cd91c2fe7b84}, intrahash = {4f0afd48875c631c876c977f03da9cc1}, keywords = {}, note = {cite arxiv:1411.6036Comment: 33 pages, 1 figure}, timestamp = {2014-12-04T01:40:42.000+0100}, title = {Discrete ABP Estimate and Convergence Rates for Linear Elliptic Equations in Non-divergence Form}, url = {http://arxiv.org/abs/1411.6036}, year = 2014 }