Overlapping Nonmatching Grid Method for the Ergodic Control Quasi Variational Inequalities

Abstract

<span style="font-family:Verdana;">In this paper, we provide a maximum norm analysis of an overlapping Schwarz method on nonmatching grids for a quasi-variational inequalities related to ergodic control problems studied by M. Boulbrach</span><span style="font-family:Verdana;">e</span><span style="font-family:Verdana;">ne [1], where the “discount factor” (</span><i><span style="font-family:Verdana;">i.e.</span></i><span style="font-family:Verdana;">, the zero order term) is set to 0, we use an overlapping Schwarz method on no</span><span style="font-family:Verdana;">n</span><span style="font-family:Verdana;">matching grid which consists in decomposing the domain in two sub domains, where the discrete alternating Schwarz sequences in sub domains converge to the solution of the ergodic control IQV for the zero order term. For </span><span style="font-family:Verdana;"> </span><img style="width:49px;height:19px;" alt="" src="Edit_ba33c230-b9bf-4532-9503-d0d7b0de6fef.bmp" width="46" height="16" /><span style="font-family:Verdana;"> </span><span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"> </span></span><span style="font-family:Verdana;">and under a discrete maximum principle we show that the discretization on each sub domain converges quasi-optimally in the </span><span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"> </span></span><span style="font-family:Verdana;">norm </span><span style="font-family:Verdana;"> </span><img alt="" src="Edit_0332f5ae-7c2d-4c2e-900b-1b5ca815c240.bmp" /><span style="font-family:Verdana;"> to 0.</span

The Maximum Norm Analysis of Schwarz Method for Elliptic Quasi-Variational Inequalities

“…also, for the non-coercive variational inequality, it has been reached in [11], the same approximation mentioned above. In [10], the authors studied a quasi-variational inequality related to control ergodic problem…”

“…(a) if the second member and the obstacle are not related to the solution, we get [2]; (b) if only the obstacle is related to the solution, we get [3]; (c) if only the second member is related to the solution, we get [4,10,11].…”