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Convergence of an alternating method to solve the Cauchy problem for Poisson’s equation. (English) Zbl 1060.35157

Let \(\Omega \in \mathbb R^2\) be a bounded open domain with sufficiently smooth boundary \(\partial \Omega\) which is decomposed in four non-empty parts \(\Gamma_0,\Gamma_1,\Gamma_2,\Gamma_3\) such that \(\partial\Omega = \Gamma_0\cap\Gamma_1\cap\Gamma_2\cap\Gamma_3\). The authors consider the Cauchy problem \[ -\Delta u = f \text{ in } \Omega, \quad u| _{\Gamma_1} =\nu_1, \quad u| _{\Gamma_2}=\nu_2, \quad \partial_\nu u| _{\Gamma_2}=g_2,\quad \partial_\nu u| _{\Gamma_3}=g_3, \] where \(f, \nu_1, \nu_2, g_2, g_3\) are known and \(\partial_\nu u\) is the outward normal derivative of \(u\). Following Kozlov et al. [V. A. Kozlov and V. G. Maz’ya, Algebra Anal. 1, No. 5, 144–170 (1989; Zbl 0732.65090), V. A. Kozlov, V. G. Maz’ya and A. V. Fomin, Zh. Vychisl. Mat. Mat. Fiz. 31, No. 1, 64–74 (1991; Zbl 0733.65056)], the authors approximate the solution of the Cauchy problem as follows: First, specify a \(\nu_0\) at \(\Gamma_0\). Then, for \(n \geq 0\), \(u^{(2n)}\) is obtained as the solution of the mixed boundary value problem \[ -\Delta u^{(2n)} = f \text{ in } \Omega, \]
\[ u^{(2n)}| _{\Gamma_0} =\nu^{(n)}, \quad u^{(2n)}| _{\Gamma_1}=\nu_1, \quad \partial_\nu u^{(2n)}| _{\Gamma_2}=g_2,\quad \partial_\nu u^{(2n)}| _{\Gamma_3}=g_3, \] where \(\nu^{(n)}\) is a function defined for all \(x \in \Gamma_0\) by \(\nu^{(0)}(x) = \nu_0| _{\Gamma_0}(x)\) and \[ \nu^{(n)}(x) = \theta u^{(2n-1)}| _{\Gamma_0}(x) + (1-\theta)\nu^{(n-1)}(x), \text{ for } n \geq 1. \] Having constructed \(u^{(2n)}\), one obtains \(u^{(2n+1)}\) by solving the mixed boundary value problem \[ -\Delta u^{(2n+1)} = f \text{ in } \Omega, \]
\[ \partial_\nu u^{(2n+1)}| _{\Gamma_0} =\partial_\nu u^{(2n)}| _{\Gamma_0}, \quad u^{(2n+1)}| _{\Gamma_1}=\nu_1, \quad u^{(2n+1)} u| _{\Gamma_2}=\nu_2,\quad \partial_\nu u^{(2n+1)}| _{\Gamma_3}=g_3. \] When \(\theta =1\), one obtains the scheme suggested by Kozlov et al. [loc. cit.] The authors prove that there is a positive constant \(\theta^* \in (1,2]\) such that, for all \(\theta \in (0,\theta^*]\) the sequence \(u^{(n)}\) converges to the solution of the Cauchy problem independently of the initial value \(\nu_0\). They present some numerical experiments based on their method. However, they do not give any theoretical result in the case when the Cauchy data are given with noise.

MSC:

35R25 Ill-posed problems for PDEs
65M30 Numerical methods for ill-posed problems for initial value and initial-boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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