In this paper we present a-posteriori error estimator for the mixed formulation of linear parabolic problem, used in designing an efficient adaptive algorithm. Our spacetime discretization consist of lowest order Raviart-Thomas finite element over graded meshes, and discontinuous Galerkin method with varying time-steps. Finally, several examples show that the proposed method is efficient and re...

In this paper, we develop and analyze a new class of discontinuous Galerkin (DG) methods for the acoustic wave equation in mixed form. Traditional mixed finite element (FE) methods produce energy conserving schemes, but these schemes are implicit, making the time-stepping inefficient. Standard DG methods give explicit schemes, but these approaches are typically dissipative or suboptimally conve...

For classical discretizations of elliptic partial differential equations, like conforming finite elements or finite differences, block Jacobi methods are equivalent to classical Schwarz methods with minimal overlap, see for example [4]. This is different when the linear system (1) is obtained using DG methods. Our paper is organized as follows: in section 2 we describe several DG methods for li...

In this paper we review a series of recent work on using a Fourier analysis technique to study the stability and error estimates for the discontinuous Galerkin method and other related schemes. The advantage of this approach is that it can reveal instability of certain “bad” schemes; it can verify stability for certain good schemes which are not easily amendable to standard finite element stabi...

A combined method consisting of the mixed finite element method for flow and the discontinuous Galerkin method for transport is introduced for the coupled system of miscible displacement problem. A “cut-off” operatorM is introduced in the discontinuous Galerkin formular in order to make the combined scheme converge. Optimal error estimates in L(H) for concentration and in L(L) for velocity are ...

In this paper we will consider the simulation of the local discontinuous Galerkin (LDG) finite element method for the porous medium equation (PME), where we present an additional nonnegativity preserving limiter to satisfy the physical nature of the PME. We also prove for the discontinuous P0 finite element that the average in each cell of the LDG solution for the PME maintains nonnegativity if...