-Error Estimates of the Extrapolated Crank-Nicolson Discontinuous Galerkin Approximations for Nonlinear Sobolev Equations

Abstract

We analyze discontinuous Galerkin methods with penalty terms, namely, symmetric interior penalty Galerkin methods, to solve nonlinear Sobolev equations. We construct finite element spaces on which we develop fully discrete approximations using extrapolated Crank-Nicolson method. We adopt an appropriate elliptic-type projection, which leads to optimal error estimates of discontinuous Galerkin approximations in both spatial direction and temporal direction.

1. Introduction

Let be an open bounded domain in ,   with smooth boundary , and let be given. In this paper, we consider the problem of approximating satisfying the following nonlinear Sobolev equations:

(1.1)

where denotes the unit outward normal vector to and is a given function defined on . The initial data , , and are assumed to be such that (1.1) admits a solution sufficiently smooth to guarantee the convergence results to be presented below. For details about the physical significance and various properties of existence and uniqueness of the Sobolev equations, see [1–6].

Early, in [7–9] the authors constructed the Galerkin approximations to the solution of (1.1) with periodic boundary conditions in one-dimensional space and obtained the optimal convergence in normed space and superconvergence results. Recently, Lin [10] constructed the Galerkin approximation of (1.1) with using Crank-Nicolson method and proved the optimal convergence of error in normed space. In [11] the authors constructed the semidiscrete finite element approximations of (1.1) with nonlinear boundary condition and obtained the optimal -error estimates.

In this work we will approximate the solution of (1.1) using a discontinuous symmetric Galerkin method with interior penalties for the spatial discretization and extrapolated Crank-Nicolson method for the time stepping. By implementing the extrapolated technique, we induce the linear systems which can be solved explicitly, and thus obviate the order reduction phenomenon which occurs when the system involved is nonlinear.

Compared to the classical Galerkin method, the discontinuous Galerkin method is very well suited for adaptive control of error and can deliver high orders of accuracy when the exact solution is sufficiently smooth. In [12] Rivière and Wheeler formulated and analyzed a family of discontinuous methods to approximate the solution of the transport problem with nonlinear reaction. They construct semidiscrete approximations which converge optimally in and suboptimally in for the energy norm and suboptimally for the norm. They also constructed fully discrete approximations and proved the optimal convergence in the temporal direction. Furthermore to solve reactive transport problems Sun and Wheeler in [13] analyzed three discontinuous Galerkin methods, namely, symmetric interior penalty Galerkin method, nonsymmetric interior penalty Galerkin method, and incomplete interior penalty Galerkin method. They obtained error estimates in which are optimal in and nearly optimal in and they developed a parabolic lift technique for SIPG which leads to -optimal and nearly -optimal error estimates in and negative norms. Recently in [14, 15] Sun and Yang adapted discontinuous Galerkin methods to nonlinear Sobolev equations and obtained the optimal error estimates. The main object of this paper is to obtain the optimal error estimates in both the spatial direction and the temporal direction by adopting an appropriate elliptic-type projection.

This paper is organized as follows. In Section 2, we introduce some notations and preliminaries. In Section 3, we construct appropriate finite element spaces and define an auxiliary projection and prove its convergence. In Section 4, we construct the extrapolated discontinuous Galerkin fully discrete method which yields the second-order convergence in the temporal direction. The corresponding error estimates of the approximate solutions are also discussed.

2. Notations and Preliminaries

Let be a regular quasi-uniform subdivision of where is a triangle or a quadrilateral if and is a -simplex or -rectangle if . Let and . Here, the regular requirement is that there exists a constant such that each contains a ball of radius . The quasi-uniformity requirement is that there is a constant such that

(2.1)

We denote the edges (resp., faces for ) of by where has positive dimensional Lebesgue measure, , , and , . With each edge (or face) , we associate a unit normal vector to if and . For , is taken to be the unit outward vector normal to .

For an and a domain , we denote by the Sobolev space of order equipped with the usual Sobolev norm . We simply write instead of if and instead of if . And also the usual seminorm defined on is denoted by .

Now for an and a given subdivision , we define the following space:

(2.2)

For with , we define the average function and the jump function such that

(2.3)

where with .

We associated the following broken norms with the space :

(2.4)

where

(2.5)

is an interior penalty term and is a discrete positive function that takes the constant value on the edge and is bounded below by and above by .

3. Finite Element Spaces and Convergence of Auxiliary Projection

For a positive integer , we construct the following finite element spaces:

(3.1)

where denotes the set of polynomials of degree less than or equal to on .

Now we state the following -approximation properties and trace inequalities whose proofs can be found in [16, 17].

Lemma 3.1.

Let and . Then there exist a positive constant depending on , , and but independent of , and and a sequence , , , such that, for any ,

(3.2)

where and is an edge or a face of .

Lemma 3.2.

For each , there exists a positive constant depending only on and such that the following trace inequalities hold:

(3.3)

where is an edge or a face of and is the unit outward normal vector to .

Now we introduce the following bilinear mappings and defined on as

(3.4)

Using the bilinear mappings and , we construct the weak formulation of problem (1.1) as follows:

(3.5)

Now for a we define the following bilinear forms and on such that

(3.6)

and satisfy the following boundedness and coercivity properties, respectively. The proofs can be found in [18, 19].

Lemma 3.3.

For a , there exists a constant satisfying

(3.7)

Lemma 3.4.

For a , there exists a constant satisfying

(3.8)

Wheeler [20] introduced an elliptic projection to prove the optimal -error estimates for Galerkin approximation to parabolic differential equations. Adopting this idea we construct a projection such that

(3.9)

By Lemmas 3.3 and 3.4, is well defined.

4. The Optimal Error Estimates of Fully Discrete Approximations

In this section we construct fully discrete discontinuous Galerkin approximations using extrapolated Crank-Nicolson method and prove the optimal convergence in normed space.

For a positive integer we let and for and we define and . For , we define ,   and .

The extrapolated Crank-Nicolson discontinuous Galerkin approximation is defined by

(4.1)

where ,  .

To apply (4.1), we need two initial stages and to be defined in the following:

(4.2)

where .

To prove the optimal convergence of in normed space we denote and , , , , .

Now we state the following approximations for whose proofs can be found in [18, 19].

Theorem 4.1.

If and then there exists a constant independent of and satisfying

(i)

(ii).

Theorem 4.2.

If , , and then there exists a constant independent of and satisfying

(i)

(ii)

provided that .

By simple computations and the applications of Theorem 4.2 we obtain the following lemmas.

Lemma 4.3.

If satisfies

(4.3)

then there exists a constant independent of and such that

(4.4)

Consequently from Lemma 4.3 there exists a constant independent of and such that

(4.5)

if is sufficiently smooth.

Lemma 4.4.

If then there exists a constant independent of and such that

(4.6)

Consequently from Lemma 4.4 there exists a constant independent of and such that

(4.7)

if is sufficiently smooth.

Lemma 4.5.

If we let then there exists a constant independent of and such that

(4.8)

Consequently from Lemma 4.5, we induce that there exists a constant independent of and such that

(4.9)

if is sufficiently smooth.

Theorem 4.6.

For and , if , then there exists a constant independent of and such that for , , ,

(4.10)

hold where and .

Proof.

From (4.1) and (1.1), we have

(4.11)

By the notations of and , we get

(4.12)

By the definition of , we obtain

(4.13)

From the definition of , we have

(4.14)

Substituting (4.12)–(4.14) in (4.11) and choosing imply that

(4.15)

By Cauchy-Schwarz'sinequality clearly we have

(4.16)

By the definition of we have

(4.17)

For the definition of we get

(4.18)

Applying (4.17) and (4.18) in (4.15) we conclude that

(4.19)

For sufficiently small by applying Lemma 4.3 there exists a constant such that

(4.20)

Applying Lemmas 4.3 and 4.4, can be estimated as follows:

(4.21)

We obtain the following estimates of for each

(4.22)

From the definition of , we can separate as follows:

(4.23)

By applying Lemma 4.5, can be estimated in the following way:

(4.24)

Similarly there exists a constant such that

(4.25)

By applying the trace inequality we have

(4.26)

From the estimation of ,  , we have

(4.27)

By applying Lemma 4.3 we obviously obtain

(4.28)

Now we can separate as follows:

(4.29)

Since

(4.30)

can be estimated as follows

(4.31)

We apply Lemma 3.2 to estimate as follows:

(4.32)

From the result of approximation of of Theorem 4.1

(4.33)

Therefore we get

(4.34)

Similarly, and are estimated as follows:

(4.35)

Substituting the estimates of , into (4.19), we get

(4.36)

If we sum both sides of (4.36) from to , then we obtain

(4.37)

which implies

(4.38)

where is sufficiently small. By applying the discrete version of Gronwall's inequality, we have

(4.39)

Therefore by applying the result of Lemma 4.7 we have

(4.40)

which proves the optimal error estimation of the fully discrete solutions.

Lemma 4.7 can be proved by the similar process of Theorem 4.6. as follows

Lemma 4.7.

For and , if , and for some constant then there exists a constant independent of and

(4.41)