Iterative two-grid methods for discontinuous Galerkin finite element approximations of semilinear elliptic problem

Abstract

In this paper, we design and analyze the iterative two-grid methods for the discontinuous Galerkin (DG) discretization of semilinear elliptic partial differential equations (PDEs). We first present an iterative two-grid method that is just like the classical iterative two-grid methods for nonsymmetric or indefinite linear elliptic PDEs, namely, to solve a semilinear problem on the coarse space and then to solve a symmetric positive definite problem on the fine space. Secondly, we designed another iterative two-grid method, which replace the semilinear term by using the corresponding first-order Taylor expansion. Specifically, we need to construct a suitable initial value, which can be sorted out from an auxiliary variational problem, for the second iterative method. We also provide the error estimates for the second iterative algorithm and present numerical experiments to illustrate the theoretical result.

Appendix

In this appendix, we present the proof of 4.24 in Lemma 4.5.

First, we define the projection operator $P_H: H^1(\Omega )\rightarrow V_H$: for $w \in H^1(\Omega )$,

$$\begin{aligned} a_h (P_H w, v_H) = a_h(w, v_H), \quad \forall v_H \in V_H. \end{aligned}$$

(2.1)

Next, we demonstrate the error estimates of $P_H$ as follows.

Lemma 2.1

For the projection operator $P_H$ and the solution $u \in H^{r+1} \cap H_0^1(\Omega )$ of 2.4, we have

$$\begin{aligned} |\Vert u - P_H u \Vert |_h \lesssim H^r \Vert u\Vert _{r+1}, \end{aligned}$$

(2.2)

$$\begin{aligned} \Vert u - P_H u\Vert _0 \lesssim H^{r+1} \Vert u\Vert _{r+1}. \end{aligned}$$

(2.3)

Proof

For any $v_H \in V_H$, using the coerciveness and continuity of $a_h(\cdot , \cdot )$ and (6.1), we could obtain that

$$\begin{aligned} |\Vert P_H u - v_H \Vert |_h^2\lesssim & {} a_h(P_H u - v_H, P_H u - v_H) \\= & {} a_h(u - v_H, P_H u - v_H) \\\lesssim & {} |\Vert u - v_H\Vert |_h |\Vert P_H u - v_H \Vert |_h, \end{aligned}$$

which implies that

$$\begin{aligned} |\Vert P_H u - v_H \Vert |_h \lesssim |\Vert u - v_H\Vert |_h . \end{aligned}$$

(2.4)

Using the triangle inequality and (6.4), we have

$$\begin{aligned} |\Vert u - P_H u\Vert |_h \le |\Vert u - v_H\Vert |_h + |\Vert v_H - P_H u\Vert |_h \lesssim |\Vert u - v_H\Vert |_h. \end{aligned}$$

(2.5)

Further using (6.5), the arbitrary of $v_H$ and 4.6, we have

$$\begin{aligned} |\Vert u - P_H u \Vert |_h \lesssim |\Vert u - \Pi _H u \Vert |_h \lesssim H^r \Vert u\Vert _{r+1}, \end{aligned}$$

which derives (6.2).

For any fixed $g \in L^2(\Omega )$, we introduce the following auxiliary problem: find $w \in H^2(\Omega ) \cap H_0^1(\Omega )$ such that

$$\begin{aligned} \left\{ \begin{aligned} - \Delta w&= g,{} & {} \text {in}\ \Omega , \\ w&= 0, \quad{} & {} \text {on}\ \partial \Omega , \end{aligned} \right. \end{aligned}$$

(2.6)

and assume that the following regularity result holds

$$\begin{aligned} \Vert w\Vert _2 \lesssim \Vert g\Vert _0. \end{aligned}$$

(2.7)

Using (6.6), similar technique in 2.8 with $w \in H^2(\Omega ) \cap H_0^1(\Omega )$, (6.1), the continuity of $a_h(\cdot , \cdot )$, 4.6 and (6.2), it is obtained that

$$\begin{aligned} \nonumber (u - P_H u, g)= & {} - (\Delta w, u - P_H u) \\ \nonumber= & {} a_h(w, u - P_H u) \\ \nonumber= & {} a_h(w - \Pi _H w, u - P_H u) \\ \nonumber\lesssim & {} |\Vert w - \Pi _H w \Vert |_h |\Vert u - P_H u\Vert |_h \\\lesssim & {} H^{r+1} \Vert w\Vert _2 \Vert u\Vert _{r+1}. \end{aligned}$$

(2.8)

At last, using (6.8), (6.7) and the arbitrary of g, we could obtain (6.3). $\square$

Assume $u \in H^{r+1}(\Omega )$ is the solution of the problem 2.4, for any $w_H \in V_H$, we define operator $\Phi _H: V_H \rightarrow V_H$ by

$$\begin{aligned} b_h(\Phi _H(w_H), v_H) = (F(u, w_H), v_H), \quad \forall \ v_H \in V_H, \end{aligned}$$

(2.9)

where

$$\begin{aligned} b_h(\Phi _H(w), v)&= a_h(\Phi _H(w), v) -(f_u(u)\Phi _H(w), v), \\ (F(u, w), v)&= (f(w), v) - (f_u(u)w, v). \end{aligned}$$

By the coerciveness and continuity of $a_h(\cdot , \cdot )$, and Assumption 2.1, we could prove that $b_h(\cdot , \cdot )$ is continuous and coercive for $V_H$, which implies the operator $\Phi _H$ is well defined.

For the given solution u of the problem 2.4, we define a space

$$\begin{aligned} B_H =\{ v_H \in V_H : \Vert P_H u - v_H\Vert _0 \le \delta _H \}, \end{aligned}$$

(2.10)

where the projection operator $P_H$ is defined in (6.1), $\delta _H = C_1 \Vert P_H u - u\Vert _0$ and $C_1$ is a constant which can be sufficiently large and does not depend on the mesh size.

Using highly similar technique in Section 2.3.4 of [14], we could prove that there is a fixed point $\bar{w}_H \in B_H$ in (6.9) by Brouwer fixed point theorem, and here is omitted. Further, we can see that the fixed point $\bar{w}_H$ is exactly the solution $\tilde{u}_H$ of 3.3, which implies $\tilde{u}_H \in B_H$. Then using the triangle inequality, (6.10) and (6.3), we have

$$\begin{aligned} \Vert \tilde{u}_H - u\Vert _0 \le \Vert \tilde{u}_H - P_H u\Vert _0 + \Vert P_H u - u \Vert _0 \lesssim \Vert P_H u - u \Vert _0 \lesssim H^{r+1} \Vert u\Vert _{r+1}, \end{aligned}$$

which completes the proof of 4.24 in Lemma 4.5.