Enforce the Dirichlet boundary condition by volume constraint in Point Integral method

Enforce the Dirichlet boundary condition by volume constraint in Point Integral method

Zuoqiang Shi Mathematical Sciences Center, Tsinghua University, Beijing, China, 100084. Email: zqshi@math.tsinghua.edu.cn.
Abstract

Recently, Shi and Sun proposed Point Integral method (PIM) to discretize Laplace-Beltrami operator on point cloud [16, 19]. In PIM, Neumann boundary is nature, but Dirichlet boundary needs some special treatment. In our previous work, we use Robin boundary to approximate Dirichlet boundary. In this paper, we introduce another approach to deal with the Dirichlet boundary condition in point integral method using the volume constraint proposed by Du et.al. [7].

1 Introduction

Partial differential equations on manifold appear in a wide range of applications such as material science [5, 9], fluid flow [12, 13], biology and biophysics [3, 10, 18, 2] and machine learning and data analysis [4, 6]. Due to the complicate geometrical structure of the manifold, it is very chanlleging to solve PDEs on manifold. In recent years, it attracts more and more attentions to develop efficient numerical method to solve PDEs on manifold. In case of that the manifold is a 2D surface embedding in , many methods were proposed include level set methods [1, 21], surface finite elements [8], finite volume methods [15], diffuse interface methods [11] and local mesh methods [14].

In this paper, we focus on following Poisson equation with Dirichlet boundary condition

(1.1)

where is a smooth manifold isometrically embedded in with the standard Euclidean metric and is the boundary. is the Laplace-Beltrami operator on manifold . Let be the Riemannian metric tensor of . Given a local coordinate system , the metric tensor can be represented by a matrix ,

Let is the inverse matrix of , then it is well known that the Laplace-Beltrami operator is

In this paper, the metric tensor is assumed to be inherited from the ambient space , that is, isometrically embedded in with the standard Euclidean metric. If is an open set in , then becomes standard Laplace operator, i.e., .

In our previous papers, [16, 19], Point Integral method was developed to solve Poisson equation in point cloud. The main observation of the Point Integral method is that the solution of the Poisson equation can be approximated by an integral equation,

(1.2)

where is the out normal of at . The kernel functions

(1.3)

and . is a parameter, which is determined by the desensity of the point cloud in the real computations.

The kernel function is assumed to be smooth and satisfies some mild conditions (see Section 1.1).

The integral approximation (1.2) is natural to solve the Poisson equation with Neumann boundary condition. To enforce the Dirichlet boundary condition, in our previous work [16, 19], we used Robin boundary condition to approximate the Dirichlet boundary condition. More specifically, we solve following problem instead of (1.1) with ,

(1.4)

Using (1.2), we have an integral equation to approximate the above Robin problem,

(1.5)

We can prove that this approach converge to the original Dirichlet problem [19]. In the real computations, small may give some trouble. The overcome this problem, we also introduced an itegrative method to enforce the Dirichlet boundary condition based on the Augmented Lagrangian Multiplier (ALM) method. However, we can not prove the convergence of this iterative method, although it always converges in the numerical tests.

Recently, Du et.al. [7] proposed volume constraint to deal with the boundary condition in the nonlocal diffusion problem. They found that in the nonlocal diffusion problem, since the operator is nonlocal, only enforce the boundary condition on the boundary is not enough, we have to extend the boundary condition to a small region close to the boundary. Borrowing this idea, in nonlocal diffusion problem to handle the Dirichlet boundary. This idea gives us following integral equation with volume constraint:

(1.6)

Here, and are subsets of which are defined as

(1.7)

The thickness of is which implies that . The relation of , , and are sketched in Fig. 1.

Figure 1: Computational domain for volume constraint

The main advantage of the integral equation (1.6) is that there is not any differential operator in the integral equation. Then it is easy to discretized on point cloud. Assume we are given a set of sample points sampling the submanifold and one vector where is the volume weight of in . In addition, we assume that the point set is a good sample of manifold in the sense that the integral on can be well approximated by the summation over , see Section 1.1.

Then, (1.6) can be easily discretized to get following linear system

(1.8)

This is the discretization of the Poisson equation (1.1) given by Point Integral Method with volume constraint on point cloud.

Similarly, the eigenvalue problem

(1.9)

can be approximated by an integral eigenvalue problem

(1.10)

And corresponding discretization is given as following

(1.11)

1.1 Assumptions and main results

One of the main contribution of this paper is that, under some assumptions, we prove that the solution of the discrete system (1.8) converges to the solution of the Poisson equation (1.1) and the spectra of the eigen problem (1.11) converge to the spectra of the Laplace-Beltrami operator with Dirichlet boundary (1.9).

The assumptions we used are listed as following.

Assumption 1.1.
  • Assumptions on the manifold: are both compact and smooth.

  • Assumptions on the sample points : is -integrable approximation of , i.e.

    • For any function , there is a constant independent of and so that

  • Assumptions on the kernel function :

    • ;

    • and for ;

    • so that for .

These assumptions are default in this paper and they are omitted in the statement of the theoretical results. And in the analysis, we always assume that and are small enough. Here, ”small enough” means that they are less than a generic constant which only depends on .

Under above assumptions, we have two theorems regarding the convergence of the Poisson equation and corresponding eigenvaule problem.

Theorem 1.1.

Let be solution of (1.1) and be solution of (1.8) and in both problems. There exists only depends on and , such that

where

(1.12)

and .

Theorem 1.2.

Let be the th largest eigenvalue of eigenvalue problem (1.9). And let be the th largest eigenvalue of discrete eigenvalue problem (1.11), then there exists a constant such that

and there exist another constant such that, for any and ,

where and , is the Riesz spectral projection associated with .

2 Stability analysis

To prove the convergence, we need some stability results which are listed in this section. The first lemma is about the coercivity of the integral operator and the proof can be found in [19].

Lemma 2.1.

For any function , there exists a constant only depends on , such that

where

and .

Next corollary directly follows from Lemma 2.1.

Corollary 2.1.

For any function , there exists a constant only depneds on , such that

where

and .

Proof.

Let

Using Lemma 2.1,

Using Lemma 2.1, we can also get following lemma regarding the stability in .

Lemma 2.2.

For any function with in , there exists a constant independent on

as long as small enough.

Proof.

Let

Since , we have

By Lemma 2.1 and the Poincare inequality, there exists a constant , such that

Let . If is smooth and close to its smoothed version , in particular,

(2.1)

then the proof is completed.

Now consider the case where does not hold. Note that we now have

Then we have

This completes the proof for the theorem. ∎

Corollary 2.2.

For any function , there exists a constant independent on , such that

as long as small enough.

Proof.

Consider

and apply Lemma 2.2. ∎

Now, we can prove one important theorem.

Theorem 2.1.

Let be solution of following integral equation

(2.2)

There exists only depends on and , such that

Proof.

First of all, we have

Now we can get estimate of . Using Corollary 2.2, we have

This gives that

(2.3)

Next, we turn to estimate the norm of in . Using the integral equation (2.2), has following expression

(2.4)

Then can be bounded as following

(2.5)

Corollary 2.1 gives a bound the first term of (2.5).

(2.6)

The second terms of (2.5) can be bounded by direct calculation.

Now we have the bound of by combining (2.5), (2.6), and (2)

(2.8)

Then the bound of can be obtained also from (2.8)

Then we have

(2.9)

The proof is completed by putting (2.3) and (2.9) together. ∎

3 Convergence analysis

The main purpose of this section is to prove that the solution of (1.8) converges to the solution of the original Poisson equation (1.1), i.e. Theorem 1.1 in Section 1.1. To prove this theorem, we split it to two parts. First, we prove that the solution of the integral equation (1.6) converges to the solution of the Poisson equation (1.1), which is given in Theorem 3.2. Then we prove Theorem 3.3 to show that the solution of (1.8) converges to the solution of (1.6).

3.1 Integral approximation of Poisson equation

To prove the convergence of the integral equation (1.6), we need following theorem about the consistency which is proved in [20].

Theorem 3.1.

Let be the solution of the problem (1.1). Let and

There exists constants depending only on and , so that for any ,

(3.1)
(3.2)

Using the consistency result, Theorem 3.1 and the stability results presented in Section 2, we can get following theorem which shows the convergence of the integral equation (1.6).

Theorem 3.2.

Let be solution of (1.1) and be solution of (1.6). There exists only depends on and , such that

Proof.

Let , first of all, we have

(3.3)

The second term can be calculated as

Here we use the definition of and the volume constraint condition to get that .

The first term is positive which is good for us. We only need to bound the second term of (3.1) to show that it can be controlled by the first term. First, the second term can be bounded as following

Here we use Lemma A.1 in Appendix A to get the last inequality.

By substituting (3.1), (3.1) in (3.3), we get

This is the key estimate we used to get convergence.

Notice that satisfying an integral equation,

(3.7)

where .

From Theorem 3.1, we know that

(3.8)
(3.9)

Now we can get estimate of . Using Corollary 2.2, we have