We present a discontinuous Galerkin method for the plate problem. The method employs a discontinuous approximation space allowing, non matching grids and diierent types of approximation spaces. Continuity is enforced weakly through the vari-ational form. Discrete approximations of the normal and twisting moments and the transversal force, which satisfy the equilibrium condition on an element le...

This paper concerns with existence, uniqueness and asymptotic behavior of the solutions for a nonlocal coupled system of reaction-diffusion. We prove the existence and uniqueness of weak solutions by the Faedo-Galerkin method and exponential decay of solutions by the classic energy method. We improve the results obtained by Chipot-Lovato and Menezes for coupled systems. A numerical scheme is pr...

Stability analyses and error estimates are carried out for a number of commonly used numerical schemes for the Allen-Cahn and Cahn-Hilliard equations. It is shown that all the schemes we considered are either unconditionally energy stable, or conditionally energy stable with reasonable stability conditions in the semi-discretized versions. Error estimates for selected schemes with a spectral-Ga...

We consider the Galerkin finite element approximation of an elliptic Dirichlet boundary control model problem governed by the Laplacian operator. The functional theoretical setting of this problem uses L2 controls and a “very weak” formulation of the state equation. However, the corresponding finite element approximation uses standard continuous trial and test functions. For this approximation,...

This paper is concerned with the stability of Petrov-Galerkin discretizations with application to parabolic evolution problems in space-time weak form. We will prove that the discrete inf-sup condition for an a priori fixed Petrov-Galerkin discretization is satisfied uniformly under standard approximation and smoothness conditions without any further coupling between the discrete trialand test ...

This paper treats the basic ideas of mixed finite element methods at an introductory level. Although the viewpoint presented is that of a mathematician, the paper is aimed at practitioners and the mathematical prerequisites are kept to a minimum. A classification of variational principles and of the corresponding weak formulations and Galerkin methods—displacement, equilibrium, and mixed—is giv...

A new numerical method based on Bernstein polynomials expansion is proposed for solving onedimensional elliptic interface problems. Both Galerkin formulation and collocation formulation are constructed to determine the expansion coefficients. In Galerkin formulation, the flux jump condition can be imposed by the weak formulation naturally. In collocation formulation, the results obtained by B-p...

The Meshless Local Petrov-Galerkin (MLPG) mixed collocation method is proposed in this paper, for solving elasticity problems. In the present MLPG approach, the mixed scheme is applied to interpolate the displacements and stresses independently, as in the MLPG finite volume method. To improve the efficiency, the local weak form is established at the nodal points, for the stresses, by using the ...

In the present paper, we deal with two different existence results of solutions for a nonlocal elliptic Dirichlet boundary value problem involving p(x)-Laplacian. The first one is based on the Brouwer fixed point theorem and the Galerkin method which gives a priori estimate of a nontrivial weak soltion. The second one is based on the variational methods. By using Mountain-Pass theorem, we obtai...