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Logarithmic convexity for discrete harmonic functions and the approximation of the Cauchy problem for Poisson’s equation. (English) Zbl 0623.65095

Consider the problem \(-\Delta u=f\), (x,y)\(\in \Omega\), \(u(0,y)=u(1,y)=0\), \(y\in [0,1]\), \(\| u-g_ 1\|_{1,\Sigma}\leq \epsilon_ 1\), \(\| u_ n-g_ 2\|_{0,\Sigma}\leq \epsilon_ 2\), \(\| u\|_{0,\Sigma '}\leq M\), where \(\Omega\) is the unit square with boundary \(\Gamma\), \(\Sigma\subset \Gamma\) and \(\Sigma\) ’\(\subset \Gamma\) are open segments on \(y=0\) and \(y=1\) respectively, \(\epsilon_ 1,\epsilon_ 2,M\geq 0\) are constants, \(f,g_ 1,g_ 2\) are given functions, \(\| \cdot \|_{m,*}\) denotes the mth Sobolev norm on the line segment *. Here the \(L_ 2\) bound on \(\Sigma\) ’ is the stabilizing supplementary condition for the ill-posed problem in interest. A numerical method for this problem is studied. For particular discrete harmonic functions defined in a finite element space of piecewise-linear functions which approximate the Laplace equation some logarithmic convexity type results are proved. Then the stability of the discrete analogue of the initial problem is obtained. Error estimates for a least squares penalty method used with discrete harmonic functions to approximate the solution are derived. Some numerical results are given.
Reviewer: S.Gocheva-Ilieva

MSC:

65M30 Numerical methods for ill-posed problems for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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