Abstract.
An implementation of the localized boundary-domain integral-equation (LBDIE) method for the numerical solution of the Neumann boundary-value problem for a second-order linear elliptic PDE with variable coefficient is discussed. The LBDIE method uses a specially constructed localized parametrix (Levi function) to reduce the BVP to a LBDIE. After employing a mesh-based discretization, the integral equation is reduced to a sparse system of linear algebraic equations that is solved numerically. Since the Neumann BVP is not unconditionally and uniquely solvable, neither is the LBDIE. Numerical implementation of the finite-dimensional perturbation approach that reduces the integral equation to an unconditionally and uniquely solvable equation, is also discussed.
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Mikhailov, S.E., Nakhova, I.S. Mesh-based numerical implementation of the localized boundary-domain integral-equation method to a variable-coefficient Neumann problem. J Eng Math 51, 251–259 (2005). https://doi.org/10.1007/s10665-004-6452-0
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DOI: https://doi.org/10.1007/s10665-004-6452-0