Weak Galerkin methods for second order elliptic interface problems

Abstract

Weak Galerkin methods refer to general finite element methods for partial differential equations (PDEs) in which differential operators are approximated by their weak forms as distributions. Such weak forms give rise to desirable flexibilities in enforcing boundary and interface conditions. A weak Galerkin finite element method (WG-FEM) is developed in this paper for solving elliptic PDEs with discontinuous coefficients and interfaces. Theoretically, it is proved that high order numerical schemes can be designed by using the WG-FEM with polynomials of high order on each element. Extensive numerical experiments have been carried out to validate the WG-FEM for solving second order elliptic interface problems. High order of convergence is numerically confirmed in both $L_2$ and $L_{\infty }$ norms for the piecewise linear WG-FEM. Special attention is paid to solve many interface problems, in which the solution possesses a certain singularity due to the nonsmoothness of the interface. A challenge in research is to design nearly second order numerical methods that work well for problems with low regularity in the solution. The best known numerical scheme in the literature is of order $\mathcal{O}(h)$ to $\mathcal{O}(h^{1.5})$ for the solution itself in $L_{\infty }$ norm. It is demonstrated that the WG-FEM of the lowest order, i.e., the piecewise constant WG-FEM, is capable of delivering numerical approximations that are of order $\mathcal{O}(h^{1.75})$ to $\mathcal{O}(h^2)$ in the $L_{\infty }$ norm for $C^1$ or Lipschitz continuous interfaces associated with a $C^1$ or $H^2$ continuous solution.

Introduction

In this paper, we are concerned with interface problems for second order elliptic partial differential equations (PDEs) with discontinuous coefficients and singular sources. For simplicity, consider the model problem of seeking functions $u=u(x\text{,}y)$ and $v=v(x\text{,}y)$ satisfying

$$-\nabla ·A_1\nabla u=f_1\text{,}\text{in}{\mathrm{\Omega }}_1\text{,}$$

$$-\nabla ·A_2\nabla v=f_2\text{,}\text{in}{\mathrm{\Omega }}_2\text{,}$$

$$u=g_1\text{,}\text{on}\partial {\mathrm{\Omega }}_1⧹\mathrm{\Gamma }\text{,}$$

$$v=g_2\text{,}\text{on}\partial {\mathrm{\Omega }}_2⧹\mathrm{\Gamma }\text{,}$$

$$u-v=\varphi \text{,}\text{on}\mathrm{\Gamma }\text{,}$$

$$A_1\nabla u·{\mathbf{n}}_1+A_2\nabla v·{\mathbf{n}}_2=\psi \text{,}\text{on}\mathrm{\Gamma }\text{,}$$

where $\mathrm{\Omega }={\mathrm{\Omega }}_1\cup {\mathrm{\Omega }}_2\text{,}\mathrm{\Gamma }={\mathrm{\Omega }}_1\cap {\mathrm{\Omega }}_2$, $\partial {\mathrm{\Omega }}_1⧹\mathrm{\Gamma }\ne \{\varnothing \}$ and ${\mathbf{n}}_1$ and ${\mathbf{n}}_2$ are outward normals of ${\mathrm{\Omega }}_1$ and ${\mathrm{\Omega }}_2$. Here, $f_1$ and $f_2$ can be singular and Γ may be of Lipschitz continuous. This problem is commonly referred to as an elliptic interface problem and occurs widely in practical applications, such as fluid mechanics [23], electromagnetic wave propagation [14], [18], [41], [40], materials science [20], [22], and biological science [39], [12], [7]. The finite difference based solution of elliptic interface problems was pioneered by Peskin with his immersed boundary method (IBM) in 1977 [30], [29]. Mayo constructed an interesting integral equation approach to this class of problems [25]. To properly solve the elliptic interface problem, one needs to enforce additional interface conditions (1.5), (1.6). LeVeque and Li advanced the subject with their second order sharp interface scheme, the immersed interface method (IIM) [24]. In the past decades, many other elegant methods have been proposed, including the ghost fluid method (GFM) proposed by Fedkiw, Osher and coworkers [11], finite-volume-based methods [28], the piecewise-polynomial discretization [8], and matched interface and boundary (MIB) methods [41], [43], [37].

A proof of second order convergence of the IIM for smooth interfaces was due to Beale and Layton [2]. Rigorous convergence analysis of most other finite difference based elliptic interface schemes is not available yet. In general, it is quite difficult to analyze the convergence of finite difference based interface schemes because conventional techniques used in Galerkin formulations are not applicable for collocation schemes. The analysis becomes particularly difficult when the designed elliptic interface scheme is capable of dealing with nonsmooth interfaces [37]. At present, there is no rigorous convergence analysis available for elliptic interface methods that delivery high-order accuracy for nonsmooth interfaces, to the authors’ knowledge.

Finite element methods (FEMs) are another class of important approaches for elliptic interface problems. The construction of FEM solutions to elliptic interface problems dates back to 1970s [1], and has been a subject of intensive investigation in the past few decades [10], [31], [17], [6]. Since the elliptic interface problem defined in Eqs. (1.1), (1.2), (1.3), (1.4), (1.5), (1.6) provides opportunities to construct new FEM schemes, a wide variety of FEM approaches have been proposed in the literature. There are two major classes of FEM based interface methods, namely, interface-fitted FEMs and immersed FEMs, categorized according to the topological relation between discrete elements and the interface. In the interface-fitted FEMs, or body-fitted FEMs, the finite element mesh is designed to align with the interface. Local mesh refinement based a priori and/or posteriori error estimation can be carried out. The performance of interface-fitted FEMs depends on the quality of the element mesh near the interface as well as the formulation of the problem. In fact, the construction of high quality FEM meshes for real world complex interface geometries is an active area of research. Bramble and King discussed an FEM for nonhomogeneous second order elliptic interface problems on smooth domains [4]. One way to deal with embedded interface conditions is to use distributed Lagrange multipliers [5], In fact, similar ideas have been widely used in mortar methods and fictitious domain methods for the treatment of embedded boundaries [13]. A $Q_1$-nonconforming finite element method was also proposed for elliptic interface problems [31]. Discontinuous Galerkin (DG) FEMs have been developed for elliptic equations with discontinuous coefficients [9], [17]. Inherited from the original DG method, DG based interface schemes have the flexibility to implement interface jump conditions.

Immersed FEMs are also effective approaches for embedded interface problems [10], [35], [15], [21], [16]. A key feature of these approaches is that their element meshes are independent of the interface geometry, i.e., the interface usually cuts through elements. As such, there is no need to use the unstructured mesh to fit the interface, and simple structured Cartesian meshes can be employed in immersed FEMs. Consequently, the time-consuming meshing process is bypassed in immersed FEMs. However, to deal with complex interface geometries, it is necessary to design appropriate interface algorithms, which are similar to finite-difference based elliptic interface methods. In fact, immersed FEMs can be regarded as the Galerkin formulations of finite difference based interface schemes. It is not surprised that key ideas of many immersed FEMs actually come from the corresponding finite-difference based interface schemes. Additionally, other ideas in numerical analysis have been utilized for the construction of immersed FEMs. For example, a stabilized Lagrange multiplier method based on Nitsche technique has been used to enforce interface jump constraints [16]. The performance of immersed FEMs depends on the design of elegant interface schemes for complex interface geometries.

Convergence analysis of FEM based elliptic interface methods has been considered by many researchers. Unlike the collocation formulation, the Galerkin framework of FEMs allows more rigorous and robust convergence analysis. Cai et al. gave a proof of convergence for a DG FEM for interface problems [6]. Dryja et al. discussed the convergence of the DG discretization of Dirichlet problems for second-order elliptic equations with discontinuous coefficients in 2D [9]. Hiptmair et al. presented a convergence analysis of $H(\text{div;}\mathrm{\Omega })$-elliptic interface problems in general 3D Lipschitz domains with smooth material interfaces [19]. Recently, an edge-based anisotropic mesh refinement algorithm has been analyzed and applied to elliptic interface problems [33].

Despite of numerous advancements in the numerical solution of interface problems, there are still a few remaining challenges in the field. One of these challenges concerns the construction of higher order interface schemes. Currently, most interface schemes are designed to be of second order convergence. However, higher order methods are efficient and desirable for problems associated with high frequency waves, such as electromagnetic and acoustic wave propagation and scattering, vibration analysis of engineering structures, and shock-vortex interaction in compressible fluid flows. It is easy to construct high order methods and even spectral methods for these problems with straight interfaces in simple domains. However, it is difficult to obtain high order convergence when the interface geometries are arbitrarily complex. A fourth order MIB scheme has been developed for the Helmholtz equation in media with arbitrarily curved interfaces in 2D [40]. Up to sixth order MIB schemes have been constructed for the Poisson equation with ellipsoidal interfaces in 3D [37]. There are two standing open problems concerning high order elliptic interface schemes, i.e., the construction of fourth-order 3D interface schemes for arbitrarily complex interfaces with sharp geometric singularities and the construction of sixth-order 3D interface schemes for arbitrarily curved smooth interfaces [37].

Another challenging issue in elliptic interface problems arises from nonsmooth interfaces or interfaces with Lipschitz continuity [39], [38], [36], [21]. Nonsmooth interfaces are also referred to as geometric singularities, such as sharp edges, cusps and tips, which commonly occur in real-world applications. It is a challenge to design high order interface schemes for geometric singularities both numerically and analytically. The first known second order accurate scheme for nonsmooth interfaces was constructed in 2007 [38]. Since then, many other interesting second order schemes have been constructed for this class of problems in 2D [8], [21], [3] and 3D [37]. The second order MIB method for 3D elliptic PDEs with arbitrarily non-smooth interfaces or geometric singularities has found its success in protein electrostatic analysis [39], [12], [7]. However, it appears truly challenging to develop fourth order schemes for arbitrarily shaped nonsmooth interfaces in 3D domains, although fourth order schemes have been reported for a few special interface geometries [37]. Due to the need in practical applications, further effort in this direction is expected.

The above mentioned difficulties in the solution of interface problems with geometric singularities significantly deteriorate in certain physical situations. It is well-known that the electric field diverges near the geometric singularities, such as tips of electrodes, antenna and elliptic cones, and sharp edges of planar conductors. Since electric field is related to the gradient of the electrostatic potential, i.e., the solution of the Poisson equation in the electrostatic analysis, it turns out that the solution of the elliptic equation has a lower regularity, e.g., the gradient does not exist at the geometric singularities. For isolated singularities, one can alleviate the difficulty by introducing an algebraic factor to solve a regularized equation whose solution has a high regularity [39], [36]. In this manner, second order MIB schemes have been constructed in the past [39], [36]. However, it is also desirable to directly solve the original PDEs with a low solution regularity. The FEM developed by Hou et al. is of 1.5th order convergence in the solution and 0.65th order convergence in the gradient of the solution when the solution of the Poisson equation is $H^2$ continuous and the interface is $C^1$ or Lipschitz continuous [21]. If the solution is only $C^1$ continuous, the convergence orders of the solution and the gradient become 1 and 0.75, respectively [21]. In fact, due to the low solution regularity induced by the nonsmooth interface, it is very difficult to analyze the convergence of numerical schemes because commonly used techniques may no longer be available. Therefore, there is a pressing need to develop both high order numerical methods and rigorous analysis of numerical methods for elliptic interface problems with low solution regularities induced by geometric singularities.

The objective of the present work is to construct a new numerical method for elliptic equations with low solution regularities induced by nonsmooth interfaces. To this end, we employ a newly developed weak Galerkin finite element method (WG-FEM) by Wang and Ye [34], [44], [45]. Like the DG methods, WG-FEM makes use of discontinuous functions in the finite element procedure which endows WG-FEMs with high flexibility to deal with geometric complexities and boundary conditions. For interface problems, such a flexibility gives rise to robustness in the enforcement of interface jump conditions. Unlike DG methods, WG-FEM enforces only weak continuity of variables naturally through well defined discrete differential operators. Therefore, weak Galerkin methods avoid pending parameters resulted from the excessive flexibility given to individual elements. As a consequence, WG-FEMs are absolutely stable once properly constructed. Recently, WG-FEMs have been applied to the solution of second-order elliptic equations [26], and the solution of Helmholtz equations with large wave numbers [27]. A major advantage of the present weak formulation is that it naturally enables WG-FEM to handle interface problems with low solution regularities. We demonstrate that the present WG-FEM of the lowest order is able to achieve from $1.75$th order to second order of convergence in the solution and about first order of convergence in the gradient when the solution of the elliptic equation is $C^1$ or $H^2$ continuous and the interface is $C^1$ or Lipschitz continuous.

The rest of this paper is organized as follows. Section 2 is devoted to a description of the method and algorithm. We shall design a weak Galerkin formulation for the elliptic interface problem given in Eqs. (1.1), (1.2), (1.3), (1.4), (1.5), (1.6). Section 3 is devoted to a convergence analysis for the weak Galerkin scheme presented in Section 2. In Section 4, we shall present some numerical results for several test cases in order to demonstrate the performance of the proposed WG-FEM for elliptic interface problems in 2D. We employ piecewise constant WG-FEMs, although piecewise linear WG-FEMs are also implemented. We first consider a few popular benchmark test examples with smooth but complex interfaces. We then carry out some investigation about nonsmooth interfaces. Some of these test problems admit solutions with low regularities, for which our numerical results significantly improve the best known result in literature and are better than our theoretical prediction. This paper ends with a conclusion, in which remarks on the use and possible advantages of high order WG-FEMs are given.