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Numerical solution of an elliptic problem with a non-classical boundary condition. (English) Zbl 1137.65422

Boyanov, Todor (ed.) et al., Numerical methods and applications. 6th international conference, NMA 2006, Borovets, Bulgaria, August 20–24, 2006. Revised papers. Berlin: Springer (ISBN 978-3-540-70940-4/pbk). Lecture Notes in Computer Science 4310, 623-627 (2007).
Summary: We investigate an elliptic problem with a boundary condition given by a sum of normal derivative and an elliptic operator in tangential variables (also known as “Venttsel” boundary condition). The differential problem is discretized by a specific finite difference method. Error estimates of the numerical method in the discrete Sobolev space \(W_2 ^1\) are obtained. The rate of convergence in this space is optimal, i.e. it is \(m - 1\) for solutions from \(W_2 ^{m}, 1 < m < 2.5\).
For the entire collection see [Zbl 1115.65003].

MSC:

65N06 Finite difference methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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