dG Finite Element Methods Research Articles

Error Approximation of the Time Dependent Hyperbolic Differential Equation by Using the DG Finite Element Method

Error Approximation of the Time Dependent Hyperbolic Differential Equation by Using the DG Finite Element Method

Error Approximation of the Second Order Hyperbolic Differential Equationby Using DG Finite Element Method

This article presents a simple efficient and asynchronously correcting a posteriori error approximation for discontinuous finite element solutions of the second-order hyperbolic partial differential problems on triangular meshes. This study considersthe basis functions for error spaces corresponding to some finite element spaces. The discretization error of each triangle is estimated by solving the local error problem. It also shows global super convergence for discontinuous solution on triangular lattice. In this article, the triangular elements are classify into three types: (i) elements with one inflow and two outflow edges are of type I, (ii) elements with two inflows and one outflow edges are of type II and (iii) elements with one inflow edge, one outflow edge, and one edge parallel to the characteristics are of type III. The article investigated higher-dimension discontinuous Galerkin methods for hyperbolic problems on triangular meshes and also studied the effect of finite element spaces on the superconvergence properties of DG solutions on three types of triangular elements and it showed that the DG solution is <i>O(h<sup>p+2</sup>)</i> superconvergent at Legendre points on the outflow edge on triangles having one outflow edge using three polynomial spaces. A posteriori error estimates are tested on a number of linear and nonlinear problems to show their efficiency and accuracy under lattice refinement for smooth and discontinuous solutions.

Analyticity and hp discontinuous Galerkin approximation of nonlinear Schrödinger eigenproblems

We study a class of nonlinear eigenvalue problems of Schrödinger type, where the potential is singular on a set of points. Such problems are widely present in physics and chemistry, and their analysis is of both theoretical and practical interest. In particular, we study the regularity of the eigenfunctions of the operators considered, and we propose and analyze the approximation of the solution via an isotropically refined [Formula: see text] discontinuous Galerkin (dG) method. We show that, for weighted analytic potentials and for up-to-quartic polynomial nonlinearities, the eigenfunctions belong to analytic-type non-homogeneous weighted Sobolev spaces. We also prove quasi optimal a priori estimates on the error of the dG finite element method; when using an isotropically refined [Formula: see text] space, the numerical solution is shown to converge with exponential rate towards the exact eigenfunction. We conclude with a series of numerical tests to validate the theoretical results.

An Error Analysis of the CN Weighed DG θ Method of the Convection Equation

In this research paper, a weighed DG finite element method is proposed for solving convection equations with an easy execution and analysis. The key aim of this method is to design an error estimation for space and time of a discontinuous approximation on general finite element meshes. The efficiency of the parameter θ in the order of convergence of the solutions is also exposed. Some numerical examples were tested that demonstrated the strength and flexibility of the method.

Treatment of Multiply Connected Domains in Time-Domain Discontinuous Galerkin $H$ –$\Phi$ Eddy Current Analysis

This paper presents a method for the treatment of multiply connected regions within the semi-explicit time-domain discontinuous Galerkin (DG) eddy current solver based on the ${H}$ – $\Phi $ field formulation. The suggested method utilizes a multivalued magnetic scalar potential over a cutting surface that divides a multiply connected domain into two simply connected regions. The multivalued magnetic scalar potential in the nonconductive domain ( $\Phi $ -domain) is regularly updated after a sequence of time steps of the explicit vector DG-finite element method solver in the electrically conductive domain ( ${H}$ -domain) yielding an accurate time-domain solution of the magnetic field in the multiply connected space surrounding the eddy current region. The efficiency and accuracy of the method are demonstrated by a 3-D example and the obtained results are verified by comparison against the corresponding frequency-domain solution.

Primal stabilized hybrid and DG finite element methods for the linear elasticity problem

Primal stabilized hybrid finite element methods for the linear elasticity problem are proposed consisting of locally discontinuous Galerkin problems in the primal variable coupled to a global problem in the multiplier which is identified with the trace of the displacement field. Numerical analysis, covering both continuous or discontinuous interpolations of the multiplier, shows that the proposed formulation preserves the main properties of the associate DG method such as consistency, stability, boundedness and optimal rates of convergence in the energy norm, and in the L2(Ω) norm for adjoint consistent formulations. Convergence studies confirm the optimal rates of convergence predicted by the numerical analysis presented here and a local post-processing technique is proposed to recover stress approximations with improved rates of convergence in H(div) norm.

Advanced Computational Engineering

The finite element method is the established simulation tool for the numerical solution of partial differential equations in many engineering problems with many mathematical developments such as mixed finite element methods (FEMs) and other nonstandard FEMs like least-squares, nonconforming, and discontinuous Galerkin (dG) FEMs. Various aspects on this plus related topics ranging from order-reduction methods to isogeometric analysis has been discussed amongst the pariticpants form mathematics and engineering for a large range of applications.

A unifying theory of a posteriori error control for discontinuous Galerkin FEM

A unified a posteriori error analysis is derived in extension of Carstensen (Numer Math 100:617–637, 2005) and Carstensen and Hu (J Numer Math 107(3):473–502, 2007) for a wide range of discontinuous Galerkin (dG) finite element methods (FEM), applied to the Laplace, Stokes, and Lame equations. Two abstract assumptions (A1) and (A2) guarantee the reliability of explicit residual-based computable error estimators. The edge jumps are recast via lifting operators to make arguments already established for nonconforming finite element methods available. The resulting reliable error estimate is applied to 16 representative dG FEMs from the literature. The estimate recovers known results as well as provides new bounds to a number of schemes.

RKDG Finite Element Method Combined with BGK Scheme for Solving Fluid Dynamics System

This paper presents a new type of second-order scheme for solving the system of Euler equations, which combines the Runge--Kutta discontinuous Galerkin (RKDG) finite element method and the Bhatnagar--Gross--Krook (BGK) scheme. We first discretize the Euler equations in space with the DG method, and then the resulting system will be discretized using an RK method. Finally, a second-order BGK method is used to construct the numerical flux. The proposed scheme keeps the main advantages of the DG finite element method, including its flexibility in handling irregular solution domains and in parallelization. Moreover, with the BGK approach the proposed schemes avoid using Riemann solvers. The efficiency and effectiveness of the proposed method are illustrated by several numerical examples in one- and two-dimensional spaces.