Time discretization of parabolic problems by the discontinuous Galerkin method (original) (raw)

Discontinuous Galerkin time discretization methods for parabolic problems with linear constraints

Journal of Numerical Mathematics, 2018

We consider time discretization methods for abstract parabolic problems with inhomogeneous linear constraints. Prototype examples that fit into the general framework are the heat equation with inhomogeneous (time-dependent) Dirichlet boundary conditions and the time-dependent Stokes equation with an inhomogeneous divergence constraint. Two common ways of treating such linear constraints, namely explicit or implicit (via Lagrange multipliers) are studied. These different treatments lead to different variational formulations of the parabolic problem. For these formulations we introduce a modification of the standard discontinuous Galerkin (DG) time discretization method in which an appropriate projection is used in the discretization of the constraint. For these discretizations (optimal) error bounds, including superconvergence results, are derived. Discretization error bounds for the Lagrange multiplier are presented. Results of experiments confirm the theoretically predicted optimal...

The discontinuous Galerkin method for semilinear parabolic problems

We prove a priori error estimates for a space-time nite element method for semilinear parabolic problems. The nite element method has basis functions that are continuous in space and discontinuous in time, and variable spatial meshes and time steps are allowed. The e ect of numerical quadrature is emphasized. R esum e. Nous montrons des estimations d'erreur a priori pour une m ethode des el ements nis en espace et en temps pour des probl emes paraboliques semi-lin eaires. La m ethode des el ements nis consider ee a des fonctions de base continues en espace et discontinues en temps, et admet des maillages spatiales et des pas de temps variables. L'e et de quadrature num erique est accentu e.

The discontinuous Galerkin method with Lax–Wendroff type time discretizations

Computer Methods in Applied Mechanics and Engineering, 2005

In this paper we develop a Lax-Wendroff time discretization procedure for the discontinuous Galerkin method (LWDG) to solve hyperbolic conservation laws. This is an alternative method for time discretization to the popular total variation diminishing (TVD) Runge-Kutta time discretizations. The LWDG is a one step, explicit, high order finite element method. The limiter is performed once every time step. As a result, LWDG is more compact than Runge-Kutta discontinuous Galerkin (RKDG) and the Lax-Wendroff time discretization procedure is more cost effective than the Runge-Kutta time discretizations for certain problems including two-dimensional Euler systems of compressible gas dynamics when nonlinear limiters are applied.

Numerical solution of parabolic integro-differential equations by the discontinuous Galerkin method

Mathematics of Computation of the American Mathematical Society, 1998

The numerical solution of a parabolic equation with memory is considered. The equation is first discretized in time by means of the discontinuous Galerkin method with piecewise constant or piecewise linear approximating functions. The analysis presented allows variable time steps which, as will be shown, can then efficiently be selected to match singularities in the solution induced by singularities in the kernel of the memory term or by nonsmooth initial data. The combination with finite element discretization in space is also studied.

Error profile for discontinuous Galerkin time stepping of parabolic PDEs

Numerical Algorithms

We consider the time discretization of a linear parabolic problem by the discontinuous Galerkin (DG) method using piecewise polynomials of degree at most r − 1 in t, for r ≥ 1 and with maximum step size k. It is well known that the spatial L2-norm of the DG error is of optimal order kr globally in time, and is, for r ≥ 2, superconvergent of order k2r− 1 at the nodes. We show that on the n th subinterval (tn− 1,tn), the dominant term in the DG error is proportional to the local right Radau polynomial of degree r. This error profile implies that the DG error is of order kr+ 1 at the right-hand Gauss–Radau quadrature points in each interval. We show that the norm of the jump in the DG solution at the left end point tn− 1 provides an accurate a posteriori estimate for the maximum error over the subinterval (tn− 1,tn). Furthermore, a simple post-processing step yields a continuous piecewise polynomial of degree r with the optimal global convergence rate of order kr+ 1. We illustrate thes...

An introduction to the discontinuous Galerkin method for convection-dominated problems

Advanced numerical approximation of nonlinear …, 1998

In these notes, we study the Runge Kutta Discontinuous Galerkin method for numericaly solving nonlinear hyperbolic systems and its extension for convectiondominated problems, the so-called Local Discontinuous Galerkin method. Examples of problems to which these methods can be applied are the Euler equations of gas dynamics, the shallow water equations, the equations of magneto-hydrodynamics, the compressible Navier-Stokes equations with high Reynolds numbers, and the equations of the hydrodynamic model for semiconductor device simulation. The main features that make the methods under consideration attractive are their formal highorder accuracy, their nonlinear stability, their high parallelizability, their ability to handle complicated geometries, and their ability to capture the discontinuities or strong gradients of the exact solution without producing spurious oscillations. The purpose of these notes is to provide a short introduction to the devising and analysis of these discontinuous Galerkin methods. Aknowledgements. The author is grateful to Al o Quarteroni for the invitation to give a series of lectures at the CIME, June 23{28, 1997, the material of which is contained in these notes. He also thanks F. Bassi and F. Rebay, and I. Lomtev and G.E. Karniadakis for kindly providing pictures from their papers 2] and 3], and 46] and 65], respectively. 1 2 Contents Preface Chapter 1. A historical overview 1.1. The original Discontinuous Galerkin method 1.2. Nonlinear hyperbolic systems: The RKDG method 1.3. Convection-di usion systems: The LDG method 1.4. The content of these notes Chapter 2. The scalar conservation law in one space dimension 2.1. Introduction 2.2. The discontinuous Galerkin-space discretization 2.3. The TVD-Runge-Kutta time discretization 2.4. The generalized slope limiter 2.5. Computational results 2.6. Concluding remarks 2.7. Appendix: Proof of the L 2-error estimates in the linear case Chapter 3. The RKDG method for multidimensional systems 3.

Aspects of discontinuous Galerkin methods for hyperbolic conservation laws

Finite Elements in …, 2002

We review several properties of the discontinuous Galerkin method for solving hyperbolic systems of conservation laws including basis construction, ux evaluation, solution limiting, adaptivity, and a posteriori error estimation. Regarding error estimation, we show that the leading term of the spatial discretization error using the discontinuous Galerkin method with degree p piecewise polynomials is proportional to a linear combination of orthogonal polynomials on each element of degrees p and p+1. These are Radau polynomials in one dimension. The discretization errors have a stronger superconvergence of order O(h 2p+1 ), where h is a mesh-spacing parameter, at the out ow boundary of each element. These results are used to construct asymptotically correct a posteriori estimates of spatial discretization errors in regions where solutions are smooth.

An hp-local Discontinuous Galerkin Method for Parabolic Integro-Differential Equations

Journal of Scientific Computing, 2010

In this article, a priori error analysis is discussed for an hp-local discontinuous Galerkin (LDG) approximation to a parabolic integro-differential equation. It is shown that the L 2-norm of the gradient and the L 2-norm of the potential are optimal in the discretizing parameter h and suboptimal in the degree of polynomial p. Due to the presence of the integral term, an introduction of an expanded mixed type Ritz-Volterra projection helps to achieve optimal estimates. Further, it is observed that a negative norm estimate of the gradient plays a crucial role in our convergence analysis. As in the elliptic case, similar results on order of convergence are established for the semidiscrete method after suitably modifying the numerical fluxes. The optimality of these theoretical results is tested in a series of numerical experiments on two dimensional domains.

Discontinuous Galerkin timestepping for nonlinear parabolic problems

PhD Thesis, University of Leicester, UK, 2018

We study space–time finite element methods for semilinear parabolic problems in (1 + d)–dimensions for d = 2, 3. The discretisation in time is based on the discontinuous Galerkin timestepping method with implicit treatment of the linear terms and either implicit or explicit multistep discretisation of the zeroth order nonlinear reaction terms. Conforming finite element methods are used for the space discretisation. For this implicit-explicit IMEX–dG family of methods, we derive a posteriori and a priori energy-type error bounds and we perform extended numerical experiments. We derive a novel hp–version a posteriori error bounds in the L1(L2) and L2(H1) norms assuming an only locally Lipschitz growth condition for the nonlinear reactions and no monotonicity of the nonlinear terms. The analysis builds upon the recent work in [60], for the respective linear problem, which is in turn based on combining the elliptic and dG reconstructions in [83, 84] and continuation argument. The a posteriori error bounds appear to be of optimal order and eÿcient in a series of numerical experiments. Secondly, we prove a novel hp–version a priori error bounds for the fully–discrete IMEX–dG timestepping schemes in the same setting in L1(L2) and L2(H1) norms. These error bounds are explicit with respect to both the temporal and spatial meshsizes kn and h, respectively, and, where possible, with respect to the possibly varying temporal polynomial degree r. The a priori error estimates are derived using the elliptic projection technique with an inf-sup argument in time. Standard tools such as Grönwall inequality and discrete stability estimates for fully discrete semilinear parabolic problems with merely locally-Lipschitz continuous nonlinear reaction terms are used. The a priori analysis extends the applicability of the results from [52] to this setting with low regularity. The results are tested by an extensive set of numerical experiments.

A posteriori error control for discontinuous Galerkin methods for parabolic problems

2008

We derive energy-norm a posteriori error bounds for an Euler timestepping method combined with various spatial discontinuous Galerkin schemes for linear parabolic problems. For accessibility, we address first the spatially semidiscrete case, and then move to the fully discrete scheme by introducing the implicit Euler timestepping. All results are presented in an abstract setting and then illustrated with particular applications. This enables the error bounds to hold for a variety of discontinuous Galerkin methods, provided that energy-norm a posteriori error bounds for the corresponding elliptic problem are available. To illustrate the method, we apply it to the interior penalty discontinuous Galerkin method, which prompts the derivation of new a posteriori error bounds. For the analysis of the time-dependent problems we use the elliptic reconstruction technique and we deal with the nonconforming part of the error by deriving appropriate computable a posteriori bounds for it.