A space–time discontinuous Galerkin method for linear convection-dominated Sobolev equations☆
Introduction
The discontinuous Galerkin method (DG method) has been studied extensively for parabolic problems by Eriksson et al. [1], Eriksson and Johnson [2], [3]. This method applies the Galerkin method in both space and time variables and thus treats space and time variables similarly. The approximate solution is sought as a polynomial in time t, with coefficients in space. The reason for considering such a method is the need for flexible schemes suitable for computations on unstructured meshes. Many works were analyzed about both the theoretical results and use of the discontinuous Galerkin method, such as [4], [5].
A space–time discontinuous Galerkin method was introduced in [6] for the nonlinear Schrödinger equation. The approach of [6] was different from that of [2], [3], [4] and based on a combination of finite element and finite difference techniques, using properties of the Radau quadrature rule [7]. In [6], optimal order error estimates in were proved under weak restrictions on the space–time meshes than that of [2], [3], [4].
Sobolev equations are one of important partial differential equations in practical use. They arise in the flow of fluids through fissured rock [8], thermodynamics [9] and other applications. For a discussion of existence and uniqueness results, see [10], [11]. For numerical treatments: semilinear problems can be found in [12]; nonlinear problems in [13], [14], [15], [16].
The purpose of this paper is to present extension of the discontinuous Galerkin method in [6] to linear Sobolev equations with convection-dominated term. More attentions are paid for treating a damping term , which is a distinct character of Sobolev equations different from parabolic equation. We present a space–time discontinuous Galerkin approximate scheme, prove the existence and uniqueness of the approximate solution and derive an optimal priori error estimate in . To our knowledge, this paper appears to be the first trial to approximate linear Sobolev equations by using the space–time discontinuous Galerkin method with Radau quadrature rule for time.
In what follows: In Section 2, we first briefly introduce linear Sobolev equations with convection-dominated term, then present a DG approximate scheme. We call this scheme as space–time DG scheme. In Section 3, we recall Radau quadrature rule and prove the existence and uniqueness of the approximate solution. In Section 4, we analyze the convergence and derive an optimal order error estimate in . In Section 5, we present results of numerical experiments, which confirm theoretical results. Finally, conclusions and perspectives are described in the last Section.
Section snippets
Sobolev equations’ space–time DG scheme
We shall consider the linear Sobolev equations: Find such thatwhere is a bounded domain in , with boundary . We define , and .
Hereafter, we will use the standard notations for Sobolev space and its norms. We assume that (2.1) admits a unique smooth solution on . For positive integer r and q used later, the regularity
Radau quadrature rule
In this section, we will prove the existence of the space–time DG scheme (2.5). The proof rests on the Radau quadrature rule, which is quoted from [7] as follows.
First for fixed integer , let be the Lagrange polynomials associated with the abscissae , i.e.
Then the Radau quadrature rule can be cited as follows: for a given function which is exact for all polynomials of degree .
Using
Some definitions
Now we turn to analyze the error. As usual, we write the error asHere W is an appropriately chosen function, which will be defined later.
Denote the elliptic projection operator by
Let be the usual Lagrange interpolation operator at the Radau points of , i.e.where are defined in (3.3). It is easy to know that and .
Numerical experiments
We provide an example showing the implementation of our algorithm. We solve Eq. (2.1) on the space–time domain , . The coefficients of Eq. (2.1) are as follows:The right-hand term is chosen so that the exact solution is (see Fig. 1)
First, we consider our space–time DG discretization on a sequence of
Conclusions and perspectives
We have presented a space–time discontinuous Galerkin method with Radau quadrature rule for time to linear Sobolev equations with convection-dominated term. More attentions have been paid for treating a damping term , which is a distinct character of Sobolev equations different from parabolic equation. We proved the existence and uniqueness of the approximate solution and derive an optimal priori error estimate in . Numerical experiments are presented to confirm theoretical
Acknowledgement
The authors thank the referees for their valuable suggestions and constructive comments.
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This research was supported by the NSF of China (Nos. 10571108, 10671113) and SRF for ROCS, SEM.