Maximum norm analysis of an overlapping nonmatching grids method for the obstacle problem

Abstract

We provide a maximum norm analysis of an overlapping Schwarz method on nonmatching grids for second-order elliptic obstacle problem. We consider a domain which is the union of two overlapping subdomains where each subdomain has its own independently generated grid. The grid points on the subdomain boundaries need not match the grid points from the other subdomain. Under a discrete maximum principle, we show that the discretization on each subdomain converges quasi-optimally in the L ∞ norm.

Overlapping Nonmatching Grid Method for the Ergodic Control Quasi Variational Inequalities

“…We begin by down a classical results related to ergodic control quasi-variational inequalities [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. It is well known that impulse control problems for reflected diffusion process may be solved by considering the solution of quasi variational inequalities (QVI) (see Bensoussan [4], A. Bensoussan and J. L. Lions [5]).…”

Finite Element Convergence Analysis of a Schwarz Alternating Method for Nonlinear Elliptic PDEs

“…For other works on finite element convergence analysis in the maximum norm of overlapping nonmatching Schwarz method, we refer to [7][8][9][10][11][12].…”

<i>L</i><sup>∞</sup>-Error Estimate of Schwarz Algorithm for Noncoercive Variational Inequalities

“…With parallel calculators, this rediscovery of these methods as algorithms of calculations was based on a modern variational approach. Pierre-Louis Lions was the starting point of an intense research activity to develop this tool of calculation, see, e.g., [1,2] and the references therein [3][4][5][6][7][8][9].…”

“…To prove the main result of this work, we construct two sequences of subsolutions and we estimate the errors between Schwarz iterates and the subsolutions. The proof stands on a Lipschitz continuous dependency with respect to the source term for variational inequality, while in [5] the proof stands on a Lipschitz continuous dependency with respect to the boundary condition.…”