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A coupling of local discontinuous Galerkin and natural boundary element method for exterior problems. (English) Zbl 1262.65172

The authors are concerned with a numerical scheme which couples local discontinuous Galerkin method with natural boundary element method in order to solve an exterior Dirichlet problem attached to Laplace equation. They claim optimal a priori error estimates in mesh dependent norms. Unfortunately, their results are based on some arbitrary and rather unclear statements. Numerical fluxes, edge-dependent penalty parameter, reduction factor with respect to the grid, etc. are just a few examples. It is also not clear at all how the numerical solution depends on the size (radius) of the truncated domain.
The paper contains some unusual formulations. The authors speak about “…boundedness and stability of the bilinear form…”. The unanimously accepted term is that of coerciveness of a bilinear form. As a matter of fact the paper is seriously polluted by numerous language flaws. See for instance p.513, “…and prove the and boundedness and stability…”, p. 525, “The results are confirmed with two numerical examples”, or p. 526, “The authors express his sincere thanks…”, to quote but a few. As it is apparent that the authors do not have contributions in the topic of coupling f. e. m. and b. e. m. the papers seems to be a simple maculated copy of some well established results.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N38 Boundary element methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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