Abstract
In this paper we address some ill-posed problems involving the heat or the wave equation in one dimension, in particular the backward heat equation and the heat/wave equation with lateral Cauchy data. The main objective is to introduce some variational mixed formulations of quasi-reversibility which enable us to solve these ill-posed problems by using some classical La-grange finite elements. The inverse obstacle problems with initial condition and lateral Cauchy data for heat/wave equation are also considered, by using an elementary level set method combined with the quasi-reversibility method. Some numerical experiments are presented to illustrate the feasibility for our strategy in all those situations. 1. Introduction. The method of quasi-reversibility has now a quite long history since the pioneering book of Latt es and Lions in 1967 [1]. The original idea of these authors was, starting from an ill-posed problem which satisfies the uniqueness property, to introduce a perturbation of such problem involving a small positive parameter e. This perturbation has essentially two effects. Firstly the perturbation transforms the initial ill-posed problem into a well-posed one for any e, secondly the solution to such problem converges to the solution (if it exists) to the initial ill-posed problem when e tends to 0. Generally, the ill-posedness in the initial problem is due to unsuitable boundary conditions. As typical examples of linear ill-posed problems one may think of the backward heat equation, that is the initial condition is replaced by a final condition, or the heat or wave equations with lateral Cauchy data, that is the usual Dirichlet or Neumann boundary condition on the boundary of the domain is replaced by a pair of Dirichlet and Neumann boundary conditions on the same subpart of the boundary, no data being prescribed on the complementary part of the boundary.
Highlights
The method of quasi-reversibility has a quite long history since the pioneering book of Lattes and Lions in 1967 [1]
One way to cope with this problem consists in using some mixed formulations of quasi-reversibility: the idea is to introduce a novel unknown which enables us to replace a fourth-order problem by two coupled second-order problems, which can be solved by Lagrange finite elements
In order to simplify the discretization with finite elements, we present a mixed formulation of quasi-reversibility to regularize the problems (1) and (2) in the H1(Q) setting
Summary
The method of quasi-reversibility has a quite long history since the pioneering book of Lattes and Lions in 1967 [1]. Heat/wave equation with lateral Cauchy data, Inverse obstacle problem, Quasi-reversibility method, Let-set method, Finite Element Method, Mixed formulation. One way to cope with this problem consists in using some mixed formulations of quasi-reversibility: the idea is to introduce a novel unknown which enables us to replace a fourth-order problem by two coupled second-order problems, which can be solved by Lagrange finite elements. We first introduce our mixed formulation of quasi-reversibility for solving the backward heat equation This formulation is adapted to the ill-posed problem of heat equation with lateral Cauchy data, with or without initial condition. We first introduce a mixed formulation to solve the wave equation with lateral Cauchy data, with or without initial condition, and secondly consider. MIXED FORMULATIONS OF QUASI-REVERSIBILITY FOR HEAT/WAVE EQUATIONS 3 the inverse obstacle problem.
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