Discontinuous Galerkin Methods for Solving Two Membranes Problem

The two membranes problem is known as a free boundary problem, which arises from a variety of applications. In this article we extend the ideas in the article [Wang, Han, and Cheng, SIAM J. Numer. Anal. 48 (2010), 703–733] to use discontinuous Galerkin methods to solve the two membranes problem. A priori error estimates are established, which reach optimal convergence order for linear elements.     

Discontinuous Galerkin methods for solving a hyperbolic inequality

In this paper, we study spatially semi‐discrete and fully discrete schemes to numerically solve a hyperbolic variational inequality, with discontinuous Galerkin (DG) discretization in space and finite difference discretization in time. Under appropriate regularity assumptions on the solution, a unified error analysis is established for four DG schemes, which reaches the optimal convergence order for linear elements. A numerical example is presented, and the numerical results confirm the theoretical error estimates.     

Upwind Discontinuous Galerkin Methods for Two Dimensional Neutron Transport Equations

In this paper the upwind discontinuous Galerkin methods with triangle meshes for two dimensional neutron transport equations will be studied. The stability for both of the semi-discrete and full-discrete method will be proved.     

Discontinuous Galerkin methods for periodic boundary value problems

This article considers the extension of well‐known discontinuous Galerkin (DG) finite element formulations to elliptic problems with periodic boundary conditions. Such problems routinely appear in a number of applications, particularly in homogenization of composite materials. We propose an approach in which the periodicity constraint is incorporated weakly in the variational formulation of the problem. Both H¹ and L² error estimates are presented. A numerical example confirming theoretical estimates is shown. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2007     

Error Estimation for Discontinuous Galerkin Solutions of Two-Dimensional Hyperbolic Problems

We analyze the discretization errors of discontinuous Galerkin solutions of steady two-dimensional hyperbolic conservation laws on unstructured meshes. We show that the leading term of the error on each element is a linear combination of orthogonal polynomials of degrees p and p+1. We further show that there is a strong superconvergence property at the outflow edge(s) of each element where the average discretization error converges as O(h ^2p+1) compared to a global rate of O(h ^p+1). Our analyses apply to both linear and nonlinear conservation laws with smooth solutions. We show how to use our theory to construct efficient and asymptotically exact a posteriori discretization error estimates and we apply these to some examples.     

Discontinuous Galerkin methods for solving double obstacle problem

In this article the ideas in Wang et al. [SIAM J Numec Anal 48 (2010), 708–73] are extended to solve the double obstacle problem using discontinuous Galerkin methods. A priori error estimates are established for these methods, which reach optimal order for linear elements. We present a test example, and the numerical results on the convergence order match the theoretical prediction. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

A New Discontinuous Galerkin Method for Parabolic Equations with Discontinuous Coefficient

In this paper, a new discontinuous Galerkin method is developed for the parabolic equation with jump coefficients satisfying the continuous flow condition. Theoretical analysis shows that this method is $L^2$ stable. When the finite element space consists of interpolative polynomials of degrees $k$, the convergent rate of the semi-discrete discontinuous Galerkin scheme has an order of$\mathcal{O}(h^k)$. Numerical examples for both 1-dimensional and 2-dimensional problems demonstrate the validity of the new method.     

Discontinuous Galerkin finite volume element methods for second‐order linear elliptic problems

In this article, a one parameter family of discontinuous Galerkin finite volume element methods for approximating the solution of a class of second‐order linear elliptic problems is discussed. Optimal error estimates in L² and broken H¹‐ norms are derived. Numerical results confirm the theoretical order of convergences. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Local Discontinuous Galerkin Methods for Three Classes of Nonlinear Wave Equations

In this paper, we further develop the local discontinuous Galerkin method to solve three classes of nonlinear wave equations formulated by the general KdV-Burgers type equations, the general fifth-order KdV type equations and the fully nonlinear K(n, n, n) equations, and prove their stability for these general classes of nonlinear equations. The schemes we present extend the previous work of Yan and Shu [30, 31] and of Levy, Shu and Yan [24] on local discontinuous Galerkin method solving partial differential equations with higher spatial derivatives. Numerical examples for nonlinear problems are shown to illustrate the accuracy and capability of the methods. The numerical experiments include stationary solitons, soliton interactions and oscillatory solitary wave solutions.The numerical experiments also include the compacton solutions of a generalized fifthorder KdV equation in which the highest order derivative term is nonlinear and the fully nonlinear K (n, n, n) equations.     

A Dynamic Frictional Contact Problem with Normal Damped Response

We consider a mathematical model which describes the frictional contact between a viscoelastic body and a reactive foundation. The process is assumed to be dynamic and the contact is modeled with a general normal damped response condition and a local friction law. We present a variational formulation of the problem and prove the existence and uniqueness of the weak solution, using results on evolution equations with monotone operators and a fixed point argument. We then introduce and study a fully discrete numerical approximation scheme of the variational problem, in terms of the velocity variable. The numerical scheme has a unique solution. We derive error estimates under additional regularity assumptions on the data and the solution.     

Domain Decomposition Preconditioners for Discontinuous Galerkin Discretizations of Compressible Fluid Flows

In this article we consider the application of Schwarz-type domain decomposition preconditioners to the discontinuous Galerkin finite element approximation of the compressible Navier-Stokes equations. To discretize this system of conservation laws, we exploit the (adjoint consistent) symmetric version of the interior penalty discontinuous Galerkin finite element method. To define the necessary coarse-level solver required for the definition of the proposed preconditioner, we exploit ideas from composite finite element methods, which allow for the definition of finite element schemes on general meshes consisting of polygonal (agglomerated) elements. The practical performance of the proposed preconditioner is demonstrated for a series of viscous test cases in both two- and three-dimensions.     

Discontinuous Galerkin finite element approximation of quasilinear elliptic boundary value problems I: the scalar case

^** Email: Paul.Houston{at}mcs.le.ac.uk^*** Email: Janice.Robson{at}comlab.ox.ac.uk^**** Email: Endre.Suli{at}comlab.ox.ac.uk We develop a one-parameter family of hp-version discontinuousGalerkin finite element methods, parameterised by [–1,1], for the numerical solution of quasilinear elliptic equationsin divergence form on a bounded open set ^d, d 2. In particular,we consider the analysis of the family for the equation –·{µ(x, |u|)u} = f(x) subject to mixed Dirichlet–Neumannboundary conditions on . It is assumed that µ is a real-valuedfunction, µ C( x [0, )), and thereexist positive constants m_µ and M_µ such that m_µ(t– s) µ(x, t)t – µ(x, s)s M_µ(t– s) for t s 0 and all x . Using a result from the theory of monotone operators for any valueof [–1, 1], the corresponding method is shown to havea unique solution u_DG in the finite element space. If u C¹() H^k(), k 2, then with discontinuous piecewise polynomials ofdegree p 1, the error between u and u_DG, measured in the brokenH¹()-norm, is (h^s–1/p^k–3/2), where 1 s min {p+ 1, k}.     

Discontinuous Galerkin derivative operators with applications to second‐order elliptic problems and stability

A discontinuous Galerkin (DG) finite‐element interior calculus is used as a common framework to describe various DG approximation methods for second‐order elliptic problems. Using the framework, symmetric interior‐penalty methods, local discontinuous Galerkin methods, and dual‐wind discontinuous Galerkin methods will be compared by expressing all of the methods in primal form. The penalty‐free nature of the dual‐wind discontinuous Galerkin method will be both motivated and used to better understand the analytic properties of the various DG methods. Consideration will be given to Neumann boundary conditions with numerical experiments that support the theoretical results. Many norm equivalencies will be derived laying the foundation for applying dual‐winding techniques to other problems. Copyright © 2015 John Wiley & Sons, Ltd.     

An $h$-Adaptive Runge-Kutta Discontinuous Galerkin Method for Hamilton-Jacobi Equations

In [35, 36], we presented an $h$-adaptive Runge-Kutta discontinuous Galerkin method using troubled-cell indicators for solving hyperbolic conservation laws. A tree data structure (binary tree in one dimension and quadtree in two dimensions) is used to aid storage and neighbor finding. Mesh adaptation is achieved by refining the troubled cells and coarsening the untroubled "children". Extensive numerical tests indicate that the proposed $h$-adaptive method is capable of saving the computational cost and enhancing the resolution near the discontinuities. In this paper, we apply this $h$-adaptive method to solve Hamilton-Jacobi equations, with an objective of enhancing the resolution near the discontinuities of the solution derivatives. One- and two-dimensional numerical examples are shown to illustrate the capability of the method.     

Discontinuous Galerkin finite element methods for variational inequalities of first and second kinds

We develop the error analysis for the h‐version of the discontinuous Galerkin finite element discretization for variational inequalities of first and second kinds. We establish an a priori error estimate for the method which is of optimal order in a mesh dependant as well as L²‐norm.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2007     

Discontinuous Galerkin method for the coupled Stokes‐Biot model

In this paper, a semi‐discrete scheme and a fully discrete scheme of the Stokes‐Biot model are proposed, and we analyze the semi‐discrete scheme in detail. First of all, we prove the existence and uniqueness of the semi‐discrete scheme, and a‐priori error estimates are derived. Then, we present the same conclusions for the fully discrete scheme. Finally, under both matching and non‐matching meshes some numerical tests are given to validate the analysis of convergence, which well support the theoretical results.     

Discontinuous Stable Elements for the Incompressible Flow

In this paper, we derive a discontinuous Galerkin finite element formulation for the Stokes equations and a group of stable elements associated with the formulation. We prove that these elements satisfy the new inf–sup condition and can be used to solve incompressible flow problems. Associated with these stable elements, optimal error estimates for the approximation of both velocity and pressure in L ² norm are obtained for the Stokes problems, as well as an optimal error estimate for the approximation of velocity in a mesh dependent norm.     

Discontinuous Galerkin finite element methods for dynamic linear solid viscoelasticity problems

We consider the usual linear elastodynamics equations augmented with evolution equations for viscoelastic internal stresses. A fully discrete approximation is defined, based on a spatially symmetric or non‐symmetric interior penalty discontinuous Galerkin finite element method, and a displacement‐velocity centred difference time discretisation. An a priori error estimate is given but only the main ideas in the proof of the error estimate are reported here due to the large number of (mostly technical) estimates that are required. The full details are referenced to a technical report. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Discontinuous Galerkin method for solving the black‐oil problem in porous media

We introduce a new algorithm for solving the three‐component three‐phase flow problem in two‐dimensional and three‐dimensional heterogeneous media. The oil and gas components can be found in the liquid and vapor phases, whereas the aqueous phase is only composed of water component. The numerical scheme employs a sequential implicit formulation discretized with discontinuous finite elements. Capillarity and gravity effects are included. The method is shown to be accurate and robust for several test problems. It has been carefully designed so that calculation of appearance and disappearance of phases does not require additional steps.     

The local discontinuous Galerkin method for the Oseen equations

We introduce and analyze the local discontinuous Galerkin method for the Oseen equations of incompressible fluid flow. For a class of shape-regular meshes with hanging nodes, we derive optimal a priori estimates for the errors in the velocity and the pressure in - and negative-order norms. Numerical experiments are presented which verify these theoretical results and show that the method performs well for a wide range of Reynolds numbers.