A posteriori error estimates for the generalized Burgers–Huxley equation with weakly singular kernels

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

Adaptive Discontinuous Galerkin Finite Element Methods for the Compressible Euler Equations

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.

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

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

The local discontinuous Galerkin finite element method for Burger’s equation

The first research topic in this thesis is the development of discontinuous Galerkin (DG) finite element methods for partial differential equations containing nonconservative products, which are present in many two-phase flow models. For this, we combine the theory of Dal Maso, LeFloch and Murat, in which a definition is given for nonconservative products even where the solution field is discontinuous. This theory also provides the mathematical foundation for a new DG finite element method. For this new DG method, we show standard (p+1)-order convergence results using p-th order basis-functions for test-cases of which we know the exact solution. We also show its ability to deal with more complex test cases. Finally, we apply the method to a depth-averaged two-phase flow model of which the numerical results are qualitatively validated against results obtained from a laboratory experiment. The second topic of this thesis is multigrid. The use of multigrid is of great importance to obtain efficient solvers for fully 3D two-phase flow models. As an initial step to improve the efficiency of solving space-time DG discretizations, we have developed, analyzed and tested optimized multigrid methods using explicit Runge-Kutta type smoothers for the 2D advection-diffusion equation. Many physical models describing fluid motion contain second (and higher) order derivatives. Obtaining a DG discretization for these higher order derivatives is non-trivial and many different DG methods exist to deal with these terms. As final topic of this thesis we introduce an alternative derivation of DG methods based on Borel measures. This alternative derivation gives a consistent treatment of derivative terms by assigning a measure to derivatives when the flow field is discontinuous. We investigate the various DG weak formulations arising from this technique by considering the 2D compressible Navier-Stokes equations for the viscous flows over a cylinder and a NACA0012 airfoil.

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 directly solving the Hamilton–Jacobi equations

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

In this paper, we develop a new discontinuous Galerkin (DG) finite element method for solving time dependent partial differential equations (PDEs) with higher order spatial derivatives. Unlike the traditional local discontinuous Galerkin (LDG) method, the method in this paper can be applied without introducing any auxiliary variables or rewriting the original equation into a larger system. Stability is ensured by a careful choice of interface numerical fluxes. The method can be designed for quite general nonlinear PDEs and we prove stability and give error estimates for a few representative classes of PDEs up to fifth order. Numerical examples show that our scheme attains the optimal $(k+1)$-th order of accuracy when using piecewise $k$-th degree polynomials, under the condition that $k+1$ is greater than or equal to the order of the equation.

An interior penalty discontinuous Galerkin finite element method for quasilinear parabolic problems

We consider a system of second-order nonlinear elliptic partial differential equations that models the equilibrium configurations of a two-dimensional planar bistable nematic liquid crystal device. Discontinuous Galerkin (dG) finite element methods are used to approximate the solutions of this nonlinear problem with nonhomogeneous Dirichlet boundary conditions. A discrete inf–sup condition demonstrates the stability of the dG discretization of a well-posed linear problem. We then establish the existence and local uniqueness of the discrete solution of the nonlinear problem. A priori error estimates in the energy and $\mathbf{L}^2$ norms are derived and a best approximation property is demonstrated. Further, we prove the quadratic convergence of the Newton iterates along with complementary numerical experiments.