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A spectral collocation method to solve Helmholtz problems with boundary conditions involving mixed tangential and normal derivatives. (English) Zbl 1055.65134
Summary: The performing spectral method, developed by {\it P. Haldenwang, G. Labrasse, S. Abboudi}, and {\it M. Deville} [J. Comput. Phys. 55, 115--128 (1984; Zbl 0544.65071)], to solve multi-dimensional Helmholtz equations, associated to mixed boundary conditions with constant coefficients, is extended to boundary conditions mixing a first order normal derivative with a second order tangential derivative. The accuracy of the proposed algorithm is evaluated on two test cases for which analytical solutions exist: an academic problem and a physical configuration including an interface with shear viscosity. The procedure is also applied to the research of the Rayleigh--Bénard instability thresholds in closed cavities with thin diffusive walls.

##### MSC:
 65N35 Spectral, collocation and related methods (BVP of PDE) 35J05 Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation 76M22 Spectral methods (fluid mechanics) 76S05 Flows in porous media; filtration; seepage
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##### References:
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