A Full Multigrid Method For Semilinear Elliptic Equation1footnote 11footnote 1This work was supported in part by National Natural Science Foundations of China (NSFC 91330202, 11371026, 11001259, 11031006, 2011CB309703) and the National Center for Mathematics and Interdisciplinary Science, CAS.

# A Full Multigrid Method For Semilinear Elliptic Equation111This work was supported in part by National Natural Science Foundations of China (NSFC 91330202, 11371026, 11001259, 11031006, 2011CB309703) and the National Center for Mathematics and Interdisciplinary Science, CAS.

Hehu Xie222LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China (hhxie@lsec.cc.ac.cn)   and  Fei Xu333Beijing Institute for Scientific and Engineering Computing, Beijing University of Technology, Beijing 100124, China (xufei@lsec.cc.ac.cn)
###### Abstract

A full multigrid finite element method is proposed for semilinear elliptic equations. The main idea is to transform the solution of the semilinear problem into a series of solutions of the corresponding linear boundary value problems on the sequence of finite element spaces and semilinear problems on a very low dimensional space. The linearized boundary value problems are solved by some multigrid iterations. Besides the multigrid iteration, all other efficient numerical methods can also serve as the linear solver for solving boundary value problems. The optimality of the computational work is also proved. Compared with the existing multigrid methods which need the bounded second order derivatives of the nonlinear term, the proposed method only needs the Lipschitz continuation in some sense of the nonlinear term.

Keywords. semilinear elliptic problem, full multigrid, multilevel correction, finite element method.

AMS subject classifications. 65N30, 65N25, 65L15, 65B99.

## 1 Introduction

The purpose of this paper is to study the multigird finite element method for semilinear elliptic problems. As we know, the multigrid and multilevel methods [3, 4, 5, 6, 9, 14, 15, 16, 21] provide optimal order algorithms for solving boundary value problems. The error bounds of the approximate solutions obtained from these efficient numerical algorithms are comparable to the theoretical bounds determined by the finite element discretization. In the past decade years, some researches about multigrid method for nonlinear elliptic problem are studied to improve the efficiency of nonlinear elliptic problem solving, i.e. [16, 22, 23]. The Newton iteration is adopted to linearize the nonlinear equation in these existing multigrid methods and then they need the bounded second order derivatives of the nonlinear terms. For more information, please refer to [10, 16, 22] and the references cited therein.

Recently, a type of multigrid method with optimal efficiency for eigenvalue problems has been proposed in [12, 17, 18, 20]. The aim of this paper is to present a full multigrid method for solving semilinear elliptic problems based on the multilevel correction scheme [17, 18]. The main idea is to design a special low dimensional space to transform the solution of the semilinear problem into a series of solutions of the corresponding linear boundary value problems on the sequence of finite element spaces and semilinear problems on a very low dimensional space. For the linearized elliptic problem, it is not necessary to solve the linear boundary value problem exactly in each correction step. Here, we only do some multigrid iteration steps for the linear boundary value problems. In this new version of multigrid method, solving semilinear elliptic problem will not be much more difficult than the multigrid scheme for the corresponding linear boundary value problems. Compared with the existing multigrid methods for the semilinear problem, our method only needs the Lipschitz continuation in some sense of the nonlinear term.

An outline of the paper goes as follows. In Section 2, we introduce the finite element method for the semilinear elliptic problem. A type of full multigrid method for the semilinear elliptic problem is given in Section 3. In Section 4, some numerical examples are provided to validate the efficiency of the proposed numerical method. Some concluding remarks are given in the last section.

## 2 Discretization by finite element method

In this paper, the letter (with or without subscripts) is used to denote a constant which may be different at different places. For convenience, the symbols , and mean that , and . Let denote a bounded convex domain with Lipschitz boundary . We use the standard notation for Sobolev spaces and their associated norms and seminorms (see, e.g. [1]). For , we denote and , where is in the sense of trace. For simplicity, we use to denote and to denote in the rest of the paper.

Here, we consider the following type of semilinear elliptic equation:

 {−∇⋅(A∇u)+f(x,u)=g,in Ω,u=0,on ∂Ω, (2.1)

where is a symmetric positive definite matrix with , is a nonlinear function with respect to the second variable.

The weak form of the semilinear problem (2.1) can be described as: Find such that

 a(u,v)+(f(x,u),v)=(g,v),∀v∈V, (2.2)

where

 a(u,v)=(A∇u,∇v). (2.3)

Obviously, is bounded and coercive on , i.e.,

 a(u,v)≤Ca∥u∥1,Ω∥v∥1,Ω   and   ca∥u∥21,Ω≤a(u,u),∀u,v∈V. (2.4)

Then we use the norm for any in this paper to replace the standard norm .

In order to guarantee the existence and uniqueness of the problem (2.2), we assume the nonlinear term satisfy the following assumption.

Assumption A: The nonlinear function satisfies the convex and Lipschitz continuous conditions as follows

 {(f(x,w)−f(x,v),w−v)≥0,   ∀w∈V, ∀v∈V,(f(x,w)−f(x,v),ϕ)≤Cf∥w−v∥0∥ϕ∥1,   ∀w∈V, ∀v∈V, ∀ϕ∈V. (2.5)

Now, we introduce the finite element method for semilinear elliptic problem (2.2). First we generate a shape regular decomposition of the computing domain into triangles or rectangles for , tetrahedrons or hexahedrons for (cf. [7, 8]). The mesh diameter describes the maximum diameter of all cells . Based on the mesh , we construct the finite element space . For simplicity, we set as the linear finite element space which is defined as follows

 Vh={vh∈C(Ω) ∣∣ vh|K∈P1,  ∀K∈Th}∩H10(Ω), (2.6)

where denotes the linear function space.

The standard finite element scheme for semilinear equation (2.2) is: Find such that

 a(¯uh,vh)+(f(x,¯uh),vh)=(g,vh),∀vh∈Vh. (2.7)

Denote a linearized operator by:

 (Lw,v)=(A∇w,∇v),  ∀w∈V, ∀v∈V.

In order to deduce the global prior error estimates, we introduce as follows:

 ηa(Vh)=supf∈L2(Ω),∥f∥0=1infvh∈Vh∥L−1f−vh∥a.

It is easy to know that as (cf. [7, 8]).

In order to measure the error for the finite element approximations, we denote

 δh(u)=infvh∈Vh∥u−vh∥a.

From [16], we can give the following error estimates.

###### Lemma 2.1.

When Assumption A is satisfied, equations (2.2) and (2.7) are uniquely solvable and the following estimates hold

 ∥u−¯uh∥a ≤ (1+Cηa(Vh))δh(u), (2.8) ∥u−¯uh∥0 ≲ ηa(Vh)∥u−¯uh∥a. (2.9)
###### Proof.

From Theorem 6.1 in [16], we can know that problems (2.2) and (2.7) are uniquely solvable. Now, it is time to prove the error estimates. For this aim, we define the finite element projection operator by the following equation

 a(Phw,vh)=a(w,vh),    ∀w∈V, ∀vh∈Vh.

It is easy to know that and . Let us define in this proof. From (2.2), (2.5) and (2.7), we have

 a(Phu−¯uh,wh) ≤ a(Phu−¯uh,wh)+(f(x,Phu)−f(x,¯uh),wh) = a(Phu,wh)+(f(x,Phu),wh)−(g,wh) = a(Phu−u,wh)+(f(x,Phu)−f(x,u),wh) = (f(x,Phu)−f(x,u),wh) ≤ Cf∥u−Phu∥0∥wh∥a.

Then the following inequalities hold

 ∥Phu−¯uh∥a≤Cf∥u−Phu∥0≤Cfηa(Vh)∥u−Phu∥a. (2.10)

Combining (2.10) and the triangle inequality leads to the following estimates

 ∥u−¯uh∥a ≤ ∥u−Phu∥a+∥Phu−¯uh∥a (2.11) ≤ δh(u)+Cfηa(Vh)∥u−Phu∥a ≤ (1+Cfηa(Vh))δh(u),

which is the desired result (2.8). From (2.10) and the triangle inequality, we have

 ∥u−¯uh∥0 ≤ ∥u−Phu∥0+∥Phu−¯uh∥0≤∥u−Phu∥0+C∥Phu−¯uh∥a ≤ Cηa(Vh)∥u−Phu∥a+Cfηa(Vh)∥u−Phu∥a ≤ (C+Cf)ηa(Vh)∥u−Phu∥a≤(C+Cf)ηa(Vh)∥u−¯uh∥a.

This is the desired result (2.9) and the proof is complete. ∎

## 3 Full multigrid method for semilinear elliptic equation

In this section, a full multigrid method for semilinear problems is proposed based on multilevel correction scheme in [17, 18]. The key point is to transform the solution of the semilinear problem into a series of solutions of the corresponding linear boundary value problems on the sequence of finite element spaces and semilinear problems on a very low dimensional space. In order to carry out the multigrid method, we first generate a coarse mesh with the mesh size and the linear finite element space is defined on the mesh . Then a sequence of triangulations of is determined as follows. Suppose (produced from by regular refinements) is given and let be obtained from via one regular refinement step (produce subelements) such that

 hk=1βhk−1,    k=2,⋯,n, (3.1)

where the positive number denotes the refinement index and larger than (always equals ). Based on this sequence of meshes, we construct the corresponding nested linear finite element spaces such that

 VH⊆Vh1⊂Vh2⊂⋯⊂Vhn. (3.2)

Due to the convexity of the domain , the sequence of finite element spaces and the finite element space have the following relations of approximation accuracy

 ηa(Vhk)≈1βηa(Vhk−1),    δhk(u)≈1βδhk−1(u),   k=2,⋯,n. (3.3)

### 3.1 One correction step

In order to design the full multigrid method, first we introduce one correction step in this subsection.

Assume we have obtained an approximate solution . A correction step to improve the accuracy of the given approximation is designed as follows.

###### Algorithm 3.1.

One Correction Step

1. Define the following auxiliary boundary value problem: Find such that

 a(ˆu(ℓ+1)hk,vhk)=−(f(x,u(ℓ)hk),vhk)+(g,vhk),∀vhk∈Vhk. (3.4)

Perform multigrid iteration steps for the second order elliptic equation to obtain an approximate solution with the following error reduction rate

 ∥˜u(ℓ+1)hk−ˆu(ℓ+1)hk∥a≤θ∥u(ℓ)hk−ˆu(ℓ+1)hk∥a, (3.5)

where is used as the initial value for the multigrid iteration and is a fixed constant independent from the mesh size .

2. Define a finite element space and solve the following semilinear elliptic equation: Find such that

 a(u(ℓ+1)hk,vH,hk)+(f(x,u(ℓ+1)hk),vH,hk)=(g,vH,hk), ∀vH,hk∈VH,hk. (3.6)

In order to simplify the notation and summarize the above two steps, we define

 u(ℓ+1)hk=SemilinearMG(VH,u(ℓ)hk,Vhk).

The error estimate of Algorithm 3.1 is studied in the following theorem.

###### Theorem 3.1.

Assume the given solution has the following estimate

 ∥¯uhk−u(ℓ)hk∥0≲ηa(VH)∥¯uhk−u(ℓ)hk∥a. (3.7)

After the one correction step defined by Algorithm 3.1, the resultant approximate solution has the following estimates

 ∥¯uhk−u(ℓ+1)hk∥a ≤ γ∥¯uhk−u(ℓ)hk∥a, (3.8) ∥¯uhk−u(ℓ+1)hk∥0 ≤ Cηa(VH)∥¯uhk−u(ℓ+1)hk∥a, (3.9)

where

 γ:=(θ+(1+θ)Cηa(VH))(1+Cηa(VH)).
###### Proof.

From (2.5), (2.7) and (3.4), we have

 a(¯uhk−ˆu(ℓ+1)hk,vhk)=(f(x,u(ℓ)hk)−f(x,¯uhk),vhk) ≤Cf∥¯uhk−u(ℓ)hk∥0∥vhk∥a≤Cηa(VH)∥¯uhk−u(ℓ)hk∥a∥vhk∥a, ∀vhk∈Vhk. (3.10)

Combing (2.4) and (3.1) leads to

 ∥¯uhk−ˆu(ℓ+1)hk∥a≤Cηa(VH)∥¯uhk−u(ℓ)hk∥a. (3.11)

After performing multigrid iteration steps, from (3.5) and (3.11), the following estimates hold

 ∥˜u(ℓ+1)hk−¯uhk∥a ≤ ∥˜u(ℓ+1)hk−ˆu(ℓ+1)hk∥a+∥ˆu(ℓ+1)hk−¯uhk∥a (3.12) ≤ θ∥u(ℓ)hk−ˆu(ℓ+1)hk∥a+∥ˆu(ℓ+1)hk−¯uhk∥a ≤ θ∥u(ℓ)hk−¯uhk∥a+θ∥ˆu(ℓ+1)hk−¯uhk∥a+∥ˆu(ℓ+1)hk−¯uhk∥a ≤ (θ+(1+θ)Cηa(VH))∥¯uhk−u(ℓ)hk∥a.

Note that the semilinear elliptic problem (3.6) can be regarded as a finite dimensional approximation of the semilinear elliptic problem (2.7). Let denotes the finite element projection operator which is defined as follows

 a(PH,hkw,vH,hk) = a(w,vH,hk),    ∀w∈V, ∀vH,hk∈VH,hk.

Since and , it is obvious that and

 ∥¯uhk−PH,hk¯uhk∥a = infvH,hk∈VH,hk∥¯uhk−vH,hk∥a (3.13) ≤ ∥¯uhk−˜u(ℓ+1)hk∥a, ∥¯uhk−PH,hk¯uhk∥0 ≤ Cηa(VH,hk)∥¯uhk−PH,hk¯uhk∥a (3.14) ≤ Cηa(VH)∥¯uhk−PH,hk¯uhk∥a.

Let us define in this proof. Based on problems (2.7) and (3.6), the following estimates hold

 a(PH,hk¯uhk−u(ℓ+1)hk,whk) (3.15) ≤ a(PH,hk¯uhk−u(ℓ+1)hk,whk)+(f(x,PH,hk¯uhk)−f(x,u(ℓ+1)hk),whk) = a(PH,hk¯uhk,wh)+(f(x,PH,hk¯uhk),whk)−(g,whk) = a(PH,hk¯uhk−¯uhk,whk)+(f(x,PH,hk¯uhk)−f(x,¯uhk),whk) = (f(x,PH,hk¯uhk)−f(x,¯uhk),whk)≤Cf∥¯uhk−PH,hk¯uhk∥0∥whk∥a.

From (3.14) and (3.15), we have

 ∥PH,hk¯uhk−u(ℓ+1)hk∥a ≤ Cf∥¯uhk−PH,hk¯uhk∥0 (3.16) ≤ Cηa(VH)∥¯uhk−PH,hk¯uhk∥a.

Combining (3.13), (3.16) and triangle inequality leads to the following inequalities

 ∥¯uhk−u(ℓ+1)hk∥a ≤ ∥¯uhk−PH,hk¯uhk∥a+∥PH,hk¯uhk−u(ℓ+1)hk∥a (3.17) ≤ (1+Cηa(VH))∥¯uhk−PH,hk¯uhk∥a ≤ (1+Cηa(VH))∥¯uhk−˜u(ℓ+1)hk∥a.

This is the desired result (3.8). From (3.15) and the triangle inequality, we have the following estimates

 ∥¯uhk−u(ℓ+1)hk∥0 ≤ ∥¯uhk−PH,hk¯uhk∥0+∥PH,hk¯uhk−u(ℓ+1)hk∥0 (3.18) ≤ ∥¯uhk−PH,hk¯uhk∥0+C∥PH,hk¯uhk−u(ℓ+1)hk∥a ≤ Cηa(VH)∥¯uhk−PH,hk¯uhk∥a ≤ Cηa(VH)∥¯uhk−u(ℓ+1)hk∥a,

which is the desired result (3.9) and the proof is complete. ∎

###### Remark 3.1.

The proof of Theorem 3.1 shows that the structure of the low dimensional space plays the key role for Algorithm 3.1. This special space makes the finite element projection has both the accuracy as in (3.13) and the -norm estimate by duality argument as in (3.14).

### 3.2 Full multigrid method

In this subsection, a full multigrid method is proposed based on the one correction step defined in Algorithm 3.1. This algorithm can reach the optimal convergence rate with the optimal computational complexity.

###### Algorithm 3.2.

Full Multigrid Scheme

1. Solve the following semilinear problem in : Find such that

 a(uh1,vh1)+(f(x,uh1),vh1)=(g,vh1),∀vh1∈Vh1.
2. For , do the following iteration:

1. Set .

2. For , do the following iterations

 u(ℓ+1)hk=SemilinearMG(VH,u(ℓ)hk,Vhk).
3. Define .

End Do

Finally, we obtain an approximate solution .

###### Theorem 3.2.

After implementing Algorithm 3.2, we have the following error estimates for the final approximation

 ∥¯uhn−uhn∥a ≤ 2γpβ1−γpβδhn(u), (3.19) ∥¯uhn−uhn∥0 ≤ Cηa(VH)∥¯uhn−uhn∥a, (3.20)

under the condition that the coarsest mesh size is small enough such that .

###### Proof.

From the first step of Algorithm 3.2, we have . Then from Lemma 2.1 and the proof of Theorem 3.1, the following estimates hold

 ∥¯uh2−uh2∥a = ∥¯uh2−u(p)h2∥a≤γp∥¯uh2−u(0)h2∥a (3.21) = γp∥¯uh2−uh1∥a=γp∥¯uh2−¯uh1∥a. ∥¯¯¯uh2−uh2∥0 ≤ Cηa(VH)∥¯uh2−uh2∥a. (3.22)

Based on (3.21), (3.22), Theorem 3.1 and recursive argument, the final approximate solution has the following error estimates

 ∥¯uhn−uhn∥a ≤ γp∥¯uhn−u(0)hn∥a=γp∥¯uhn−uhn−1∥a ≤ γp(∥¯uhn−¯uhn−1∥a+∥¯uhn−1−uhn−1∥a) ≤ γp∥¯uhn−¯uhn−1∥a+γ2p(∥¯uhn−1−¯uhn−2∥a+∥¯uhn−2−uhn−2∥a) ≤ n−1∑k=1γkp∥¯uhn−k+1−¯uhn−k∥a ≤ n−1∑k=1γkp(∥¯uhn−k+1−u∥a+∥u−¯uhn−k∥a) ≤ 2n−1∑k=1γkpδhn−k(u)≤2n−1∑k=1γkpβkδhn(u)≤2γpβ1−γpβδhn(u),

which is just the desired result (3.19). The second result (3.20) can be proved by the similar argument in the proof of Theorem 3.1 and the proof is complete. ∎

###### Corollary 3.1.

For the final approximation obtained by Algorithm 3.2, we have the following estimates

 ∥u−uhn∥a ≲ δhn(u), (3.23) ∥u−uhn∥0 ≲ ηa(VH)δhn(u). (3.24)
###### Proof.

This is a direct consequence of the combination of Lemma 2.1 and Theorem 3.2. ∎

### 3.3 Estimate of the computational work

In this subsection, we turn our attention to the estimate of computational work for the full multigrid method defined in Algorithm 3.2. It will be shown that the full multigrid method makes solving the semilinear elliptic problem almost as cheap as solving the corresponding linear boundary value problems.

First, we define the dimension of each level finite element space as . Then we have

 Nk≈(1β)d(n−k)Nn,   k=1,2,⋯,n. (3.25)

The computational work for the second step in Algorithm 3.2 is different from the linear elliptic problems [4, 14, 15, 16, 21]. In this step, we need to solve a semilinear elliptic problem (3.6). Always, some type of nonlinear iteration method (fixed-point iteration or Newton type iteration) is adopted to solve this low dimensional semilinear elliptic problem. In each nonlinear iteration step, it is required to assemble the matrix on the finite element space () which needs the computational work . Fortunately, the matrix assembling can be carried out by the parallel way easily in the finite element space since it has no data transfer.

###### Theorem 3.3.

Assume we use computing-nodes in Algorithm 3.2, the semilinear elliptic solving in the coarse spaces () and need work and , respectively, and the work of the multigrid iteration for the boundary value problem in each level space is for . Let denote the nonlinear iteration times when we solve the semilinear elliptic problem (3.6). Then in each computational node, the work involved in Algorithm 3.2 has the following estimate

 Total work = O((1+ϖϑ)Nn+MHlogNn+Mh1). (3.26)
###### Proof.

We use to denote the work involved in each correction step on the -th finite element space . From the definition of Algorithm 3.2, we have the following estimate

 Wk = O(Nk+MH+ϖNkϑ). (3.27)

Based on the property (3.25), iterating (3.27) leads to

 Total work = (3.28) = O(n∑k=2(1+ϖϑ)Nk+(n−1)MH+Mh1) = O(n∑k=2(1β)d(n−k)(1+ϖϑ)Nn+MHlogNn+Mh1) = O((1+ϖϑ)Nn+MHlogNn+Mh1).

This is the desired result and we complete the proof. ∎

###### Remark 3.2.

Since we always have a good enough initial solution in the second step of Algorithm 3.1, then solving the semilinear elliptic problem (3.6) never needs many nonlinear iterations. In this case, the complexity in each computational node will be provided and