Convergence rates for the quasi-reversibility method to solve the Cauchy problem for Laplace's equation

Abstract

We consider the quasi-reversibility method to solve the Cauchy problem for Laplace's equation in a smooth bounded domain. We assume that the Cauchy data are contaminated by some noise of amplitude σ, so that we make a regular choice of ε as a function of σ, where ε is the small parameter of the quasi-reversibility method. Specifically, we present two different results concerning the convergence rate of the solution of quasi-reversibility to the exact solution when σ tends to 0. The first result is a convergence rate of type 1/(log 1 σ ) β in a truncated domain, the second one holds when a source condition is assumed and is a convergence rate of type σ 1 2 in the whole domain. The Cauchy problem for Laplace's equationLet Ω be an open set of R N , N ≥ 2, which satisfies the following assumptions, referred by (A) from now on. Assumptions (A): Ω is bounded, connected and of class C 2 [3]. Furthermore, Γ 0 and Γ 1 are two open subsets of ∂Ω with mes(Γ 0 ) > 0 and mes(Γ 1 ) > 0, Γ 0 ∩ Γ 1 = ∅ and Γ 0 ∪ Γ 1 = ∂Ω.

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