Abstract
An unfitted discontinuous Galerkin method is proposed for the elliptic interface problems. Based on a variant of the local discontinuous Galerkin method, we obtain the optimal convergence for the exact solution u in the energy norm and its flux p in the L2 norm. These results are the same as those in the case of elliptic problems without interface. Finally, some numerical experiments are presented to verify our theoretical results.
1. Introduction
Elliptic interface problems are often encountered in many multiphysics and multiphase applications in science computing and engineering. For example, second order elliptic equations with discontinuous coefficients are often used to model problems in material sciences and fluid dynamics when two or more distinct materials or fluids with different conductivities, densities, or permeability are involved. It is well known that, when the interface is smooth enough, the solution of elliptic interface problems has higher regularity in individual material or fluid region than in the entire physical domain.
To numerically solve such interface problems, first we need to generate a mesh. One approach is to use a body fitted mesh. However, for those problems where the interface moves with time, repeated remeshing of the domain to obtain a fitted mesh is very costly. Another one is to use an unfitted grid independent of the location of the interface. This technique is particularly preferred to simulate time-dependent problems with moving interfaces. The major advantage for using an unfitted mesh is that it avoids repeatedly remeshing the domain for fitting the moving interfaces.
As for fitted mesh method for elliptic problems with interface, Chen and Zou in [1] considered the finite element method for solving elliptic and parabolic interface problems, and almost-optimal error estimates in the norm and energy norm were obtained. In [2], the authors studied a class of discontinuous Galerkin method for elliptic interface problems, which was shown to be optimally convergent in norm. Recently, a high-order HDG method was presented to solve elliptic interface problems by Huynh et al. in [3], which was extended to solve Stokes interface flow in [4].
Various unfitted grid methods for interface problems have been proposed in the literature. Finite difference methods are very popular unfitted grid methods due to their simplicity, for example, the immersed interface method [5, 6], the immersed boundary method [7], the boundary condition capturing method [8], and many others. And there exist many works for finite element methods on unfitted grid as well. Babuška in [9] studied the elliptic interface problem on unfitted mesh and derived suboptimal convergence behavior. Li et al. proposed an immersed interface finite element method in [10], which modified the basis functions near interface to satisfy the homogeneous jump conditions. Later, this method was applied to elliptic and elasticity interface problems with nonhomogeneous jump conditions in [11, 12]. Recently, an unfitted finite element method based on Nitsche’s method was presented by A. Hansbo and P. Hansbo in [13] and optimal order of convergence was proved without restrictions on the location of the interface relative to the mesh, which was used to solve incompressible elasticity with discontinuous modulus in [14]. More recently, Massjung in [15] considered an unfitted discontinuous Galerkin method which was viewed as a generalization of Hansbo’s method in [13]. An optimal convergence rate with respect toand a suboptimal convergence rate with respect toin energy norm were proved. Later, Wu and Xiao also presented an unfitted interface penalty finite element method, which was extended to the three dimensional case in [16].
The local discontinuous Galerkin (LDG) method was proposed by Cockburn and Shu in [17] to solve general time-dependent convection-diffusion problems. Later, the method was carried to elliptic problems for mixed discontinuous Galerkin formulation by Castillo et al. in [18]. The purpose of this paper is to extend the LDG method to a class of elliptic problems with a smooth interface. However, employing an unfitted mesh method, the interface can divide regular grid cells into degenerated subcells. If this situation happens, the standard inverse estimates can no longer be valid. In this paper, we use the weighted average instead of the arithmetic average in the classic LDG method to retrieve the inverse estimates (see Lemmas 8 and 10). Thus, we propose an unfitted discontinuous Galerkin method, based on a variant of LDG method. We prove the optimal convergence rate of the method for the exact solutionin the energy norm and its fluxin thenorm, respectively.
The rest of this paper is organized as follows. In Section 2 we propose our DG method and present some necessary preliminaries. We prove optimal order error estimates for our DG method in Section 3. In Section 4, some numerical experiments are presented to justify our theoretical results. Finally, conclusions are given in Section 5.
Let us now end this section with some notation to be used in this paper. We will use the standard notations for Sobolev spaces and norms in this paper (see [19, 20]). In particular, for a bounded open setin, if and , we denote bythe Sobolev space of functionssuch that, wheredenotes the standard Sobolev space with norm . As usual we define the broken norm:. Throughout the paper, the generic constantis always independent of the mesh parameter.
2. Discontinuous Galerkin Method and Preliminaries
Letbe a bounded domain in with convex polygonal boundaryandan open domain withboundary . Let (see Figure 1). We consider the following elliptic interface problem: whereis the outward pointing unit normal toandis the jump ofacross the interface, whereis the restrictions ofon. For the sake of simplicity, we assume that the coefficientis a positive and piecewise constant; that is, .
Regarding the regularity for the solution of the interface problem (1), we state without proof the following theorem.
Theorem 1 (cf. [1]). Assume that and . Then problem (1) has a unique solution , and the following a priori estimate holds:
By introducing the flux , the interface problem (1) can be rewritten into a first order system as
Let be a shape regular and locally quasi-uniform simplicial triangulation of , generated independently of the location of the interface. For the definition of shape regular and locally quasi-uniform, we refer to [20, 21]. Suppose to be made of straight triangles with diameter . As usual, let . The set of edges of the triangulation is denoted by , and is the set of interior edges of . For any element, denote the part ofin by; that is, . For any edge , let . We call the elements whose interiors are cut through by “interface elements”, and denote the set of the interface elements by . For an interface element , assume that is the part of interface intersecting . For the geometrical features of the interface , we give the following plausible assumptions (cf. [13, 15]).
Assumption 2. We assume that intersects the boundary of an element exactly twice and each (open) edge at most once.
Assumption 3. Let be the straight line segment connecting the points of intersection between and . We assume that is a function of length on , in local coordinates:
Assumption 4. Suppose that with ; then there exist triangles and such that and
To formulate our numerical scheme, first we define two usual discontinuous finite element spaces as where denotes the space of polynomials of degree less than or equal to on each element.
We define our discontinuous finite element spaces as
Following the notation of [22], let be an interior edge shared by two triangles and in . For a scalar valued function , piecewise smooth on with , we define the jump and the weighted average ofas Similarly, for a vector valued function , piecewise smooth on with , we set where is the unit normal of pointing towards the outside of and , . If is an edge on the boundary of , we define on where denotes the unit outer normal of pointing towards the outside of .
For the weight average across interfaceof any piecewise smooth function discontinuous on , we set where whose specific definitions will be given in Lemma 10.
For simplicity, for and , we define
Following [18], testing the problem (3) by and , respectively, using integration by parts and noting the identities and , we obtain our DG method: find , such that for all , where is the stabilization parameter.
We define the bilinear and linear forms And integral by parts yields that
Hence, our DG approximation can be written as the following mixed variational problem: find , such that For the exactof the interface problem (1) and, using Theorem 1, we have , on , and, across . Then the following consistency property holds:
Let the mesh-dependent norm be defined by
Theorem 5. Suppose that the stabilization parameter is positive; then the DG method (16) defines a unique approximate solution .
Proof. Since (16) is a square system, it is enough to show uniqueness. Let , . Setting and, adding the two equations of (16), we have which deduces, on and across . As a consequence, the first equation of (16) becomes Hence, taking implies . Since on and across , we conclude that . This completes the proof.
The following lemma comes from the famous Stein’s extension theorem.
Lemma 6 (cf. [19]). There exist two extension operatorsfor all nonnegative integers such that where .
Next, we state a standard approximation lemma.
Lemma 7 (cf. [20, 21]). Let . Then forthere exists a linear continuous operatorsuch that
The following two lemmas are variant inverse estimates involving interface which play important role in our analysis.
Lemma 8 (cf. [15]). For let ; then for the following inverse inequality holds:
Lemma 9 (cf. [15, 16]). The following estimate holds for either or , for any :
By Lemma 9, we can immediately obtain the following result.
Lemma 10. Let ; then there exists a positive constant such that where
Lemma 11. Letand; then we have
Proof. Under Assumptions 2 and 3, the trace inequality (29) follows from Lemma 3 in [13] and a scaling argument.
3. Error Estimate of Our DG Method
Now, we define interpolation operator by and . Let and . To obtain the convergence result, we need to show the following approximate error estimates.
Lemma 12. Suppose that and ; then the following approximate estimates hold:
Proof. We only need to show inequality (30); inequality (31) can be shown similarly. For the first term on the left-hand side of (30), by the inequality (22) from Lemma 7 we obtain Summing over all triangles, it follows by Lemma 6 that Next, we estimate the second term on the left-hand side of (30) as follows. Suppose that with ; using the inequality (23) from Lemma 7 yields Summing over all edges, using Lemma 6 implies that Similarly, due to the inequality (24) from Lemma 7 and Lemmas 6 and 11, we find Thus the inequality (30) follows combining (33)–(36), which completes the proof.
Next we present a priori error estimate of the exact solution in the energy norm and its fluxin the-norm.
Theorem 13. Let be the solution of (3) and the solution of (16), respectively. Then, for , the following error estimate holds:
Proof. By using the consistency property (17), we have
Set , in (38) to give
Using (31) in Lemma 12 and -Cauchy-Schwartz inequality, we estimate the first term on the right-hand side of (39) as
For the second term on the right-hand side of (39), by using (30) in Lemma 12 and trace inequalities from Lemmas 8 and 10, we obtain
Similarly, by using the approximate estimation (31), the third term on (39) can be estimated as
Then, by virtue of inequality (30) in Lemma 12, we bound the fourth term on (39) as
Thus, combining (39)–(43) yields
By using the triangle inequality, we have
At the other hand, setting in the first equality of (38), we obtain
By the definition of , an integration by parts implies that
Using Lemmas 8 and 10 yields
Combining (45) and (48), choosingenough small, from the definition and the triangle inequality, we can arrive at
which completes the proof.
Using the standard duality argument, we can obtain the following error estimate in the norm.
Theorem 14. Under the condition of Theorem 13, we have
Proof. Consider the following so-called adjoint problem Using Theorem 1, we get By introducing an auxiliary variable , we obtain Since on and on , using integration by parts and the consistency property (17), we deduce Due to , as in the proof of Theorem 13, an application of Theorem 1 implies that Now we estimate as follows: Similarly, We now combine (54)–(57) and use (52) and Theorem 13 to obtain which completes the proof.
4. Numerical Experiments
In this section, we consider the following nontrivial example with a homogeneous jump condition (from [10, 13]). The exact solution is given by where , and on the domain we choose . A simple calculation shows that .
We compute the order of convergence for and , when piecewise linear polynomials are used to approximateand , respectively. In Figure 2, we plot the numerical solution for this example on a uniform mesh for 1 : 1000. Figures 3, 4, 5, and 6 show the computed order of convergence for and when the jump in the coefficient is taken as 1 : 10, 1 : 1000, 10 : 1, 1000 : 1, respectively, in the log-log scale. These computed results coincide with the theoretical results in Theorems 13 and 14.
5. Conclusions
In this paper, we have discussed an unfitted discontinuous Galerkin method for elliptic problems with a smooth interface. Based on a variant of local discontinuous Galerkin method, we have obtained the optimal order error estimates in the energy norm inand its flux . And by using the standard duality argument the optimal convergence rate in norm for has also been derived. These presented results are the same as that of elliptic problems without interface. Finally, numerical experiments are given to confirm our theoretical results. We note that the convergence behavior in most existing works concerning the elliptic interface problems depends on the jump in the discontinuous coefficients. It will be one of our future subjects to design an efficient numerical scheme that is robust with respect to the jump in the discontinuous coefficients.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors would like to express sincere gratitude to the reviewers for their constructive suggestions which helped improve the quality of this paper. This work was supported by the National Science Foundation (NSF) of China (Grants no. 11371199 and 11301275), the Program of Natural Science Research of Jiangsu Higher Education Institutions of China (Grant no. 12KJB110013), and the Doctoral Fund of Ministry of Education of China (Grant no. 20123207120001).