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Paredes, Diego; Valentin, Frédéric; Versieux, Henrique M.

Revisiting the robustness of the multiscale hybrid-mixed method: the face-based strategy. (English) Zbl 1522.65223

J. Comput. Appl. Math. 436, Article ID 115415, 18 p. (2024).
Summary: This work proposes a new finite element for the multiscale hybrid-mixed method (MHM) applied to the Poisson equation with highly oscillatory coefficients. Unlike the original MHM method, multiscale bases are the solution to local Neumann problems driven by piecewise continuous polynomial interpolation on the skeleton faces of the macroscale mesh. As a result, we prove the optimal convergence of MHM by refining the face partition and leaving the mesh of macroelements fixed. This property allows the MHM method to be resonance free under the usual assumptions of local regularity. The numerical analysis of the method also revisits and complements the original approach proposed by D. Paredes et al. [Math. Comput. 86, No. 304, 525–548 (2017; Zbl 1355.65159)]. Numerical experiments assess the new theoretical results.

Cited in 1 Document

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
35B65 Smoothness and regularity of solutions to PDEs
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs

Keywords:

multiscale methods; FEM; elliptic equations

Citations:

Zbl 1355.65159
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Full Text: DOI HAL

References:

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