DEFINITION AND ANALYSIS OF A WAVELET/FICTITIOUS DOMAIN SOLVER FOR THE 2D-HEAT EQUATION ON A GENERAL DOMAIN

Abstract

This paper is devoted to the setting and analysis of a Petrov–Galerkin wavelet-fictitious domain numerical method for the approximation of the solution of multi-dimensional parabolic equations on a general domain. In this method, the original parabolic equation, set on a domain ω, is first discretized in time using a finite difference scheme. At each time step, the corresponding elliptic equation on ω is transformed into a saddle point problem on a functional space defined on a bigger but simply shaped domain Ω where the initial boundary conditions on ω are enforced using surface Lagrange multipliers (Ref. 2). The solution of this problem is then approximated, thanks to a Petrov–Galerkin formulation, using wavelets and time scheme associated "vaguelettes" (Ref. 8). Existence, uniqueness and convergence of the approximated solution, when finite dimension spaces are used, are established and the efficiency as well as the stability of the numerical algorithm (namely the Uzawa algorithm) used in the resolution are analyzed. The constraint of a discrete inf-sup condition as well as ill-conditioning associated to the trace operator are investigated in the wavelet framework. Numerical results related to the 2D heat equation are presented.