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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