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A non-local boundary value problem method for the Cauchy problem for elliptic equations. (English) Zbl 1170.35555

Let \(H\) be a Hilbert space with norm \(\|\cdot\|\), \(A:D(A)\subset H\to H\) a positive definite, self-adjoint operator with compact inverse on \(H\), and \(T\) and \(\varepsilon\) given positive numbers. The ill-posed Cauchy problem for elliptic equations
\[ u_{tt}= Au, \quad 0< t< T, \qquad \|u(0)-\varphi\|\leq\varepsilon, \qquad u_t(0)=0, \]
is regularized by the well-posed non-local boundary value problem
\[ u_{tt}= Au, \quad 0< t< aT, \qquad \|u(0)+\alpha u(aT)=\varphi, \qquad u_t(0)=0, \]
with \(a\geq1\) being given and \(\alpha>0\) the regularization parameter. A priori and a posteriori parameter choice rules are suggested which yield order-optimal regularization methods. Numerical results based on the boundary element method are presented and discussed.

MSC:

35R25 Ill-posed problems for PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
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