In this paper we propose and analyze a Discontinuous Galerkin method for a linear parabolic problem with dynamic boundary conditions. We present the formulation and prove stability and optimal a priori error estimates for the fully discrete scheme. More precisely, using polynomials of degree p ≥ 1 on meshes with granularity h along with a backward Euler time-stepping scheme with time-step ∆t, w...

We present a discrete analysis of non-reflecting boundary conditions for the discontinuous Galerkin method. The boundary conditions considered in this paper include the recently proposed Perfectly Matched Layer absorbing boundary condition for the linearized Euler equation and two non-reflecting boundary conditions based on the characteristic decomposition of the flux on the boundary. The analy...

We present a detailed study of spatially propagating waves in a discontinuous Galerkin scheme applied to a system of linear hyperbolic equations. We start with an eigensolution analysis of the semi-discrete system in one space dimension with uniform grids. It is found that, for any given order of the basis functions, there are at most two spatially propagating numerical wave modes for each phys...

This paper presents the lowest-order weak Galerkin finite element method for solving the Darcy equation on quadrilateral and hybrid meshes consisting of quadrilaterals and triangles. In this approach, the pressure is approximated by constants in element interiors and on edges. The discrete weak gradients of these constant basis functions are specified in local Raviart-Thomas spaces RT[0] for qu...

We consider the continuous space-time Galerkin method for the linear second-order wave equation proposed by French and Peterson in 1996. A bottleneck for this approach is how to solve the discrete problems effectively. In this paper, we tackle this bottleneck by essentially employing wavelet bases in space. We show how to decouple the corresponding linear system and we prove that the resulting ...

In this paper, we investigate numerical schemes for solving a three component CahnHilliard model. The space discretization is performed by using a Galerkin formulation and the finite element method. For the time discretization, the main difficulty is to write a scheme ensuring, at the discrete level, the decrease of the energy. We study three different schemes and propose existence and converge...

Abstract. This article is devoted to the study of the finite element approximation for a nonlocal nonlinear parabolic problem. Using a linearised Crank-Nicolson Galerkin finite element method for a nonlinear reaction-diffusion equation, we establish the convergence and error bound for the fully discrete scheme. Moreover, important results on exponential decay and vanishing of the solutions in f...

We introduce and analyze two new semi-discrete numerical methods for the multi-dimensional Vlasov-Poisson system. The schemes are constructed by combing a discontinuous Galerkin approximation to the Vlasov equation together with a mixed finite element method for the Poisson problem. We show optimal error estimates in the case of smooth compactly supported initial data. We propose a scheme that ...

In this paper, we study the semi-discrete Galerkin finite element method for parabolic equations with Lipschitz continuous coefficients. We prove the maximumnorm stability of the semigroup generated by the corresponding elliptic finite element operator, and prove the space-time stability of the parabolic projection onto the finite element space in L∞(QT ) and L p((0, T ); Lq ( )), 1 < p, q < ∞....