A Variational Method and Approximations of a Cauchy Problem for Elliptic Equations

Abstract

A Cauchy problem for general elliptic second-order linear partial differential equations in which the Dirichlet data in H1/2(Γ1 U Γ3) is assumed available on a larger part of the boundary Γ of the bounded domain Ω than the boundary portion Γ1 on which the Neumann data is is investigated using a conjugate gradient method. We obtain an approximation to the solution of the Cauchy problem by minimizing a certain discrete functional and interpolating using the finite diference or boundary element method. The minimization involves solving equations obtained by discretising mixed boundary value problems for the same operator and its adjoint. It is proved that the solution of the discretised optimization problem converges to the continuous one, as the mesh size tends to zero. Numerical results are presented and discussed.