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Pre-Publicación 2019-15

Ricardo Oyarzúa, Manuel Solano, Paulo Zuñiga:

A priori and a posteriori error analyses of a high order unfi tted mixed-FEM for Stokes flow

Abstract:

We propose and analyze a high order unfitted mixed finite element method for the pseudostress-velocity formulation of the Stokes problem with Dirichlet boundary condition on a fluid domain Ω with curved boundary Γ. The method consists of approximating Ω by a polygonal subdomain D_h, with boundary Γ_h, where a Galerkin method is applied to approximate the solution, and on a transferring technique, based on integrating the extrapolated discrete gradient of the velocity, to approximate the Dirichlet boundary data on the computational boundary Γ_h. The associated Galerkin scheme is defined by Raviart–Thomas elements of order k≥0 for the pseudostress and discontinuous polynomials of degree k for the velocity. Provided suitable hypotheses on the mesh near the boundary Γ, we prove well-posedness of the Galerkin scheme by means of the Babuška–Brezzi theory and establish the corresponding optimal convergence O(h^(k+1)). Moreover, for the case when Γ_h is constructed through a piecewise linear interpolation of Γ, we propose a reliable and quasi-efficient residual-based a posteriori error estimator. Its definition make use of a postprocessed velocity with enhanced accuracy to achieve the same rate of convergence of the method when the solution is smooth enough. Numerical experiments illustrate the performance of the scheme, show the behaviour of the associated adaptive algorithm and validate the theory.

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