The paper is concerned with the discontinuous Galerkin finite element method for the numerical solution of nonlinear conservation laws and nonlinear convection-diffusion problems with emphasis on applications to the simulation of compressible flows. We discuss two versions of this method: (a) Finite volume discontinuous Galerkin method, which is a generalization of the combined finite volume—fi...

We summarize several techniques of analysis for finite element methods for linear hyperbolic problems, illustrating their key properties on the simplest model problem. These include the discontinuous Galerkin method, the continuous Galerkin methods on rectangles and triangles, and a nonconforming linear finite element on a special triangular mesh.

We consider a discontinuous Galerkin finite element method for the advection–reaction equation in two space–dimensions. For polynomial approximation spaces of degree greater than or equal to two on triangles we propose a method where stability is obtained by a penalization of only the upper portion of the polynomial spectrum of the jump of the solution over element edges. We prove stability in ...

A family of explicit space-time finite element methods for the initial boundary value problem for linear, symmetric hyperbolic systems of equations is described and analyzed. The method generalizes the discontinuous Galerkin method and, as is typical for this method, obtains error estimates of order O(hn+1/2) for approximations by polynomials of degree ≤ n.

In this paper an implicit fully discrete local discontinuous Galerkin (LDG) finite element method is applied to solve the time-fractional seventh-order Korteweg-de Vries (sKdV) equation, which is introduced by replacing the integer-order time derivatives with fractional derivatives. We prove that our scheme is unconditional stable and L error estimate for the linear case with the convergence ra...

This work presents the coupling of two locally conservative methods for elliptic problems: namely, the discontinuous Galerkin method and the mixed finite element method. The couplings can be defined with or without interface Lagrange multipliers. The formulations are shown to be equivalent. Optimal error estimates are given; penalty terms may or may not be included. In addition, the analysis fo...

Aalto University, P.O. Box 11000, FI-00076 Aalto www.aalto.fi Author Matti Leinonen Name of the doctoral dissertation On applying stochastic Galerkin finite element method to electrical impedance tomography Publisher School of Science Unit Department of Mathematics and Systems Analysis Series Aalto University publication series DOCTORAL DISSERTATIONS 135/2015 Field of research Mathematics Manus...

(1) A one-dimensional minimization problem and the Ritz method. (2) Weak formulation and the Galerkin method. Abstract error estimates. (3) The lemma of variations. Euler-Lagrange equations. Weak formulation, again. (4) The Ritz-Galerkin finite element method: philosophy. (5) The piecewise linear finite element space and basis functions. The linear system. (6) The piecewise linear finite elemen...

(2001) A coupled Element Free Galerkin / Boundary Element method for stress analysis of two-dimensional solids. Abstract Element Free Galerkin (EFG) method is a newly developed meshless method for solving partial differential equations using Moving Least Squares interpolants. It is, however, computationally expensive for many problems. A coupled EFG/Boundary Element (BE) method is proposed in t...

We present a higher order accurate discontinuous Galerkin finite element method for the simulation of linear free-surface gravity waves. The method uses the classical Runge-Kutta method for the time discretization of the free-surface equations and the discontinuous Galerkin method for the space discretization. In order to circumvent numerical instabilities arising from an asymmetric mesh a stab...