An anisotropic a priori error analysis for a convection-dominated diffusion problem using the HDG method

This paper deals with the a priori error analysis for a convection-dominated diffusion 2D problem, when applying the HDG method on a family of anisotropic triangulations. It is known that in this case, boundary or interior layers may appear. Therefore, it is important to resolve these layers in order to recover, if possible, the expected order of approximation. In this work, we extend the use of HDG method on anisotropic meshes. To this end, some assumptions need to be asked to the stabilization parameter, as well as to the family of triangulations. In this context, when the discrete local spaces are polynomials of degree k ≥ 0, this approach is able to recover an order of convergence k + 1 2 in L 2 for all the variables. Numerical examples confirm our theoretical results.Fil: Bustinza, Rommel. Universidad de Concepción; ChileFil: Lombardi, Ariel Luis. Universidad Nacional de Rosario. Facultad de Ciencias Exactas Ingeniería y Agrimensura. Escuela de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; ArgentinaFil: Solano, Manuel. Universidad de Concepción; Chil

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

An a-priori error analysis for discontinuous Lagrangian finite elements applied to nonconforming dual-mixed formulations: Poisson and Stokes problems. ETNA - Electronic Transactions on Numerical Analysis

In this paper, we discuss the well-posedness of a mixed discontinuous Galerkin (DG) scheme for the Poisson and Stokes problems in 2D, considering only piecewise Lagrangian finite elements. The complication here lies in the fact that the classical Babuška-Brezzi theory is difficult to verify for low-order finite elements, so we proceed in a non-standard way. First, we prove uniqueness, and then we apply a discrete version of Fredholm's alternative theorem to ensure existence. The a-priori error analysis is done by introducing suitable projections of the exact solution. As a result, we prove that the method is convergent, and, under standard additional regularity assumptions on the exact solution, the optimal rate of convergence of the method is guaranteed

A mixed Hybrid High-Order formulation for linear interior transmission elliptic problems

In this paper we devise a new Hybrid High-Order (HHO) method for a linear elliptic transmission problem in a bounded domain. In HHO the solution of the problem at hand is approximated by attaching polynomials of degree k to the mesh cells and to their boundaries. Specific element-local operators are then employed to obtain a high-order reconstruction of the solution. Following this construction, a well-posed nonconforming discrete formulation is obtained. A significant advantage of HHO is that cell-based unknowns can be eliminated locally via a Schur complement, obtaining a global problem posed on the mesh skeleton. This in turn allows to obtain a compact global linear system with a significantly reduced number of unknowns. In our scheme an auxiliary unknown, which plays the role of a Lagrange multiplier, is introduced to deal with the nonhomogeneous transmission conditions. We prove that the proposed method is optimally convergent in the energy norm, as well as in the L 2-norm for the potential and a weighted L 2-norm for the Lagrange multiplier, for smooth enough solutions. Finally, we include some numerical experiments that validate our theoretical results, even in situations not covered by the current analysis

A mixed Hybrid High-Order formulation for linear interior transmission elliptic problems

In this paper we devise a new Hybrid High-Order (HHO) method for a linear elliptic transmission problem in a bounded domain. In HHO the solution of the problem at hand is approximated by attaching polynomials of degree k to the mesh cells and to their boundaries. Specific element-local operators are then employed to obtain a high-order reconstruction of the solution. Following this construction, a well-posed nonconforming discrete formulation is obtained. A significant advantage of HHO is that cell-based unknowns can be eliminated locally via a Schur complement, obtaining a global problem posed on the mesh skeleton. This in turn allows to obtain a compact global linear system with a significantly reduced number of unknowns. In our scheme an auxiliary unknown, which plays the role of a Lagrange multiplier, is introduced to deal with the nonhomogeneous transmission conditions. We prove that the proposed method is optimally convergent in the energy norm, as well as in the L 2-norm for the potential and a weighted L 2-norm for the Lagrange multiplier, for smooth enough solutions. Finally, we include some numerical experiments that validate our theoretical results, even in situations not covered by the current analysis

A look at how LDG and BEM can be coupled

In this work we explain the coupling of Local Discontinuous Galerkin and general Boundary Element methods for a class of nonlinear exterior transmission problems. The method is exposed as the discrete coupling of two Neumann–to–Dirichlet operators, one coping with all nonlinearities and source terms and the other for a purely exterior Laplace equation. We show how two separate methods for interior and exterior problems can be coupled at a weak level using a mortar space and how the analysis can be organised with very little interaction of the interior and exterior parts by considering the natural seminorms to measure the influence of the coupling unknown on the different methods

Enriched finite element subspaces for dual–dual mixed formulations in fluid mechanics and elasticity

[Abstract] In this paper we unify the derivation of finite element subspaces guaranteeing unique solvability and stability of the Galerkin schemes for a new class of dual-mixed variational formulations. The approach, which has been applied to several linear and nonlinear boundary value problems, is based on the introduction of additional unknowns given by the flux and the gradient of velocity, and by the stress and strain tensors and rotations, for fluid mechanics and elasticity problems, respectively. In this way, the procedure yields twofold saddle point operator equations as the resulting weak formulations (also named dual–dual ones), which are analyzed by means of a slight generalization of the well known Babuška–Brezzi theory. Then, in order to introduce well posed Galerkin schemes, we extend the arguments used in the continuous case to the discrete one, and show that some usual finite elements need to be suitably enriched, depending on the nature of the problem. This leads to piecewise constant functions, Raviart–Thomas of lowest order, PEERS elements, and the deviators of them, as the appropriate subspaces