A Fast Discontinuous Galerkin Finite Element Method for a Bond-Based Linear Peridynamic Model

Discontinuous Galerkin finite element method for solving population density functions of cortical pyramidal and thalamic neuronal populations

This paper presents three new coupling methods for interior penalty discontinuous Galerkin finite element methods and boundary element methods. The new methods allow one to use discontinuous basis functions on the interface between the subdomains represented by the finite element and boundary element methods. This feature is particularly important when discontinuous Galerkin finite element methods are used. Error and stability analysis is presented for one of the methods. Numerical examples suggest that all three methods exhibit very similar convergence properties, consistent with available theoretical results.

A comparative study on the weak Galerkin, discontinuous Galerkin, and mixed finite element methods

Discontinuous Galerkin finite element (DGFE) methods offer a mathematically beautiful, computationally efficient, and efficiently parallelizable way to solve partial differential equations (PDEs). These properties make them highly desirable for numerical calculations in relativistic astrophysics and many other fields. The BSSN formulation of the Einstein equations has repeatedly demonstrated its robustness. The formulation is not only stable but allows for puncture-type evolutions of black hole systems. To-date no one has been able to solve the full (3 + 1)-dimensional BSSN equations using DGFE methods. This is partly because DGFE discretization often occurs at the level of the equations, not the derivative operator, and partly because DGFE methods are traditionally formulated for manifestly flux-conservative systems. By discretizing the derivative operator, we generalize a particular flavor of DGFE methods, Local DG methods, to solve arbitrary second-order hyperbolic equations. Because we discretize at the level of the derivative operator, our method can be interpreted as either a DGFE method or as a finite differences stencil with non-constant coefficients.

Four-order superconvergent CDG finite elements for the biharmonic equation on triangular meshes

Analysis of two discontinuous Galerkin finite element methods for the total pressure formulation of linear poroelasticity model

The stress-strain constitutive law for viscoelastic materials such as soft tissues, metals at high temperature, and polymers can be written as a Volterra integral equation of the second kind with a fading memory kernel. This integral relationship yields current stress for a given strain history and can be used in the momentum balance law to derive a mathematical model for the resulting deformation. We consider such a dynamic linear viscoelastic model problem resulting from using a Dirichlet–Prony series of decaying exponentials to provide the fading memory in the Volterra kernel. We introduce two types of internal variable to replace the Volterra integral with a system of auxiliary ordinary differential equations and then use a spatially discontinuous symmetric interior penalty Galerkin (SIPG) finite element method and – in time – a Crank–Nicolson method to formulate the fully discrete problems: one for each type of internal variable. We present a priori stability and error analyses without using Grönwall’s inequality and with the result that the constants in our estimates grow linearly with time rather than exponentially. In this sense, the schemes are therefore suited to simulating long time viscoelastic response, and this (to our knowledge) is the first time that such high quality estimates have been presented for SIPG finite element approximation of dynamic viscoelasticity problems. We also carry out a number of numerical experiments using the FEniCS environment (https://fenicsproject.org), describe a simulation using “real” material data, and explain how the codes can be obtained and all of the results reproduced.

A numerical methodology is proposed to discretize a nonlinear low-frequency approximation to Maxwell’s equations using a local discontinuous Galerkin (DG) finite element method, with an upwind-like numerical flux, for modeling superconductors. In this paper, we focus on high-temperature superconductors (HTS) and the electrical resistivity is modeled using a power law. Nodal elements and the Whitney element are used. Numerical studies have been performed to verify the proposed methodology: a problem with a manufactured solution, the nonlinear magnetic front problem, and the magnetization of HTS wires. Based on the final time that can be reached for a given time-step size, the proposed strategy is compared with the $\mathbf {H}$ formulation discretized using the Galerkin finite element method with the Whitney element for the magnetic front problem. The proposed local DG strategy allows the use of a larger time-step size over a longer time interval, particularly, when we use the Whitney element. The proposed methodology can also capture sharp gradients of the current density with limited spurious oscillations. The numerical results are in agreement with Bean’s model for large values of power-law’s exponent. The proposed local DG strategy could be generalized to more complex electrical resistivity models, including multiphysics models.

In this paper we develop and analyze the local discontinuous Galerkin (LDG) finite element method for solving the general Lax equation. The local discontinuous Galerkin method has the flexibility for arbitrary h and p adaptivity, and allows for hanging nodes. By choosing the numerical fluxes carefully we prove stability and give an error estimate. Finally some numerical examples are computed to show the convergence order and excellent numerical performance of proposed method.

A discontinuous Galerkin finite element method for solving the Troesch’s problem

Space–time discontinuous Galerkin finite element method for two-fluid flows

The local discontinuous Galerkin finite element method for a class of convection–diffusion equations

Application of Runge-Kutta Discontinuous Galerkin finite element method to shallow water flow

The direct discontinuous Galerkin (DDG) finite element method, using piecewise polynomials of degree k ≥ 1 on a Shishkin mesh, is applied to convection-dominated singularly perturbed two-point boundary value problems. Consistency, stability and convergence of order k (up to a logarithmic factor) are proved in an energy-type norm appropriate to the method and problem. The results are robust, i.e., they hold uniformly for all values of the singular perturbation parameter. Numerical experiments confirm the theoretical convergence rate.

We present an entropy stable Discontinuous Galerkin (DG) finite element method to approximate systems of 2-dimensional symmetrizable conservation laws on unstructured grids. The scheme is constructed using a combination of entropy conservative fluxes and entropy-stable numerical dissipation operators. The method is designed to work on structured as well as on unstructured meshes. As solutions of hyperbolic conservation laws can develop discontinuities (shocks) in finite time, we include a multidimensional slope limitation step to suppress spurious oscillations in the vicinity of shocks. The numerical scheme has two steps: the first step is a finite element calculation which includes calculations of fluxes across the edges of the elements using 1-D entropy stable solver. The second step is a procedure of stabilization through a truly multi-dimensional slope limiter. We compared the Entropy Stable Scheme (ESS) versus Roe’s solvers associated with entropy corrections and Osher’s solver. The method is illustrated by computing solution of the two stationary problems: a regular shock reflection problem and a 2-D flow around a double ellipse at high Mach number.