Fully discrete FEM-BEM method for a class of exterior nonlinear parabolic-elliptic problems in 2D

[Abstract] We considered a nonlinear parabolic equation in a bounded domain of R2 coupled with the Laplace equation in the corresponding exterior region. This kind of problems appears in the modelling of quasi-stationary electromagnetic fields. We chose a regular artificial boundary containing the nonlinear region in its interior. Then, we applied a symmetric FEM-BEM coupling procedure including a parameterization of the artificial boundary. We used the backward Euler method for the time discretization and an exact triangulation of the finite element domain. Assuming that the nonlinear operator is strongly monotone and Lipschitz-continuous, we proved convergence and obtained optimal error estimates for the solution of the discrete problem. Finally, we proposed a fully discrete scheme with quadrature formulas of low order and, under some additional conditions on the nonlinearity, proved that the order of convergence remains optimal

A fully discrete BEM-FEM method for an exterior elasticity system in the plane

The final publication is available http://dx.doi.org/10.1016/S0377-0427(00)00533-1[Abstract] We present a modified version of the usual BEM–FEM coupling for the exterior elasticity problem in the plane (cf. Costabel and Stephan, SIAM J. Numer. Anal. 27 (1990) 1212–1226). This new formulation allows us to take advantage of techniques from Hsiao et al., Computing 25 (1980) 557–566 to compute the boundary integral terms using simple quadrature formulas. We provide error estimates for the Galerkin method and prove that the corresponding fully discrete scheme preserves the optimal rates of convergen

On the Adaptive Numerical Solution to the Darcy–Forchheimer Model †

Presented at the 4th XoveTIC Conference, A Coruña, Spain, 7–8 October 2021.[Abstract] We considered a primal-mixed method for the Darcy–Forchheimer boundary value problem. This model arises in fluid mechanics through porous media at high velocities. We developed an a posteriori error analysis of residual type and derived a simple a posteriori error indicator. We proved that this indicator is reliable and locally efficient. We show a numerical experiment that confirms the theoretical results.The authors acknowledge the support of CITIC (FEDER Program and grant ED431G 2019/01). The research of M.G. is partially supported by Xunta de Galicia Grant GRC ED431C 2018-033. The research of H.V. is partially supported by Ministerio de Educación grant FPU18/06125.Xunta de Galicia; ED431G 2019/01Xunta de Galicia; ED431C 2018-03

Numerical Simulation of a Nonlinear Problem Arising in Heat Transfer and Magnetostatics

[Abstract] We present a numerical model that comprises a nonlinear partial differential equation. We apply an adaptive stabilised mixed finite element method based on an a posteriori error indicator derived for this particular problem. We describe the numerical algorithm and some numerical results.The authors acknowledge the support of CITIC (FEDER Program and grant ED431G 2019/01). The research of M.G. is partially supported by the Spanish Ministerio de Economía y Competitividad Grant MTM2016-76497-R, and by Xunta de Galicia Grant GRC ED431C 2018-033. The research of H.V. is partially supported by Xunta de Galicia grant ED481A-2019/413170 and Ministerio de Educación grant FPU18/06125Xunta de Galicia; ED431G 2019/01Xunta de Galicia; GRC ED431C 2018-033Xunta de Galicia; ED481A-2019/41317

On a FEM--BEM formulation for an exterior quasilinear problem in the plane

The publication is available http://dx.doi.org/10.1137/S0036142998335364[Abstract] We use a version of the FEM--BEM method introduced by Costabel [ Boundary Elements IX, Vol. 1, C. A. Brebbia et al., eds., Springer-Verlag, 1987] and Han [ J. Comput. Math., 8 (1990), pp. 223--232] to discretize an exterior quasilinear problem. We provide error estimates for the Galerkin method and propose a fully discrete scheme based on simple quadrature formulas. Furthermore, we show that these numerical integration schemes preserve the optimal rates of convergence. Finally, we present results of numerical experiments involving our discretization method

A low-order mixed finite element method for a class of quasi-Newtonian Stokes flows. Part I: a priori error analysis

[Abstract] We present a mixed finite element method for a class of non-linear Stokes models arising in quasi-Newtonian fluids. Our results include, as a by-product, a new mixed scheme for the linear Stokes equation. The approach is based on the introduction of both the flux and the tensor gradient of the velocity as further unknowns, which yields a twofold saddle point operator equation as the resulting variational formulation. We prove that the continuous and discrete formulations are well posed, and derive the associated a priori error analysis. The corresponding Galerkin scheme is defined by using piecewise constant functions and Raviart–Thomas spaces of lowest order

A mixed finite element method for the generalized Stokes problem

[Abstract] We present and analyse a new mixed finite element method for the generalized Stokes problem. The approach, which is a natural extension of a previous procedure applied to quasi-Newtonian Stokes flows, is based on the introduction of the flux and the tensor gradient of the velocity as further unknowns. This yields a two-fold saddle point operator equation as the resulting variational formulation. Then, applying a slight generalization of the well known Babuška–Brezzi theory, we prove that the continuous and discrete formulations are well posed, and derive the associated a priori error analysis. In particular, the finite element subspaces providing stability coincide with those employed for the usual Stokes flows except for one of them that needs to be suitably enriched. We also develop an a posteriori error estimate (based on local problems) and propose the associated adaptive algorithm to compute the finite element solutions. Several numerical results illustrate the performance of the method and its capability to localize boundary layers, inner layers, and singularities

A posteriori error analysis of an augmented mixed formulation in linear elasticity with mixed and Dirichlet boundary conditions

[Abstract] We develop a residual-based a posteriori error analysis for the augmented mixed methods introduced in and for the problem of linear elasticity in the plane. We prove that the proposed a posteriori error estimators are both reliable and efficient. Numerical experiments confirm these theoretical properties and illustrate the ability of the corresponding adaptive algorithms to localize the singularities and large stress regions of the solutions