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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 $$u_{tt}= Au, \quad 0< t< T, \qquad \|u(0)-\varphi\|\le\varepsilon, \qquad u_t(0)=0,$$ 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 $a\ge1$ 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.

35R25Improperly posed problems for PDE
65M60Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (IVP of PDE)
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