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Finite element methods for Darcy’s problem coupled with the heat equation. (English) Zbl 1393.35159

Summary: In this article, we study theoretically and numerically the heat equation coupled with Darcy’s law by a nonlinear viscosity depending on the temperature. We establish existence of a solution by using a Galerkin method and we prove uniqueness. We propose and analyze two numerical schemes based on finite element methods. An optimal a priori error estimate is then derived for each numerical scheme. Numerical experiments are presented that confirm the theoretical accuracy of the discretization.

MSC:

35Q35 PDEs in connection with fluid mechanics
35K05 Heat equation
76S05 Flows in porous media; filtration; seepage
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
35B45 A priori estimates in context of PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65H10 Numerical computation of solutions to systems of equations
65N15 Error bounds for boundary value problems involving PDEs

Software:

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Full Text: DOI

References:

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