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

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.

[1]  I. Daubechies,et al.  Biorthogonal bases of compactly supported wavelets , 1992 .

[2]  G. Chiavassa,et al.  On the Effective Construction of Compactly Supported Wavelets Satisfying Homogenous Boundary Conditions on the Interval. , 1997 .

[3]  Jaroslav Haslinger,et al.  FICTITIOUS DOMAINS METHODS WITH DISTRIBUTED LAGRANGE MULTIPLIERS PART I: APPLICATION TO THE SOLUTION OF ELLIPTIC STATE PROBLEMS , 2001 .

[4]  Angela Kunoth,et al.  Wavelet Techniques for the Fictitious-Domain-Lagrange-Multiplier-Approach , 2001, Numerical Algorithms.

[5]  Faker Ben Belgacem,et al.  The Mortar finite element method with Lagrange multipliers , 1999, Numerische Mathematik.

[6]  Ingrid Daubechies,et al.  Ten Lectures on Wavelets , 1992 .

[7]  R. Glowinski,et al.  Error analysis of a fictitious domain method applied to a Dirichlet problem , 1995 .

[8]  Richard Pasquetti,et al.  A Spectral Embedding Method Applied to the Advection-Diffusion Equation , 1996 .

[9]  Claudio Canuto,et al.  The wavelet element method. Part I: Construction and analysis. , 1997 .

[10]  Albert Cohen,et al.  Wavelet methods in numerical analysis , 2000 .

[11]  Guillaume Chiavassa,et al.  A fully adaptive wavelet algorithm for parabolic partial differential equations ? ? This work has be , 2001 .

[12]  Charles K. Chui,et al.  An Introduction to Wavelets , 1992 .

[13]  Wolfgang Dahmen,et al.  Appending boundary conditions by Lagrange multipliers: Analysis of the LBB condition , 2001, Numerische Mathematik.

[14]  Philippe Angot,et al.  A penalization method to take into account obstacles in incompressible viscous flows , 1999, Numerische Mathematik.