Optimal error bound for immersed weak Galerkin finite element method for elliptic interface problems

doi:10.1016/j.cam.2022.114567

Journal of Computational and Applied Mathematics

Volume 416, 15 December 2022, 114567

https://doi.org/10.1016/j.cam.2022.114567 Get rights and content

Abstract

In Mu and Zhang (2019), an immersed weak Galerkin finite element method (IWG-FEM) is developed for solving elliptic interface problems and it is proved that this method has optimal a-priori error estimate in an energy norm under artificial smoothness assumption on the solution. In this study, we prove that IWG-FEM converges optimally in energy norm under natural smoothness assumption on solution. Furthermore, we show that IWG-FEM converges optimally in the $L^{2}$ norm which did not present in Mu and Zhang (2019) because of the artificial $H^{3}$ smoothness requirement. A series of numerical experiments are conducted and reported to verify the theoretical finding.

Introduction

Many applied problems involve interfaces, such as Stokes flow with moving interfaces [1], [2], Stefan problems [3], [4], Hele-Shaw flow [5], bubble simulations in incompressible two-phase flow [6], elasticity problems with interfaces in material science [7], [8], [9], problems of multiple phase elastic materials separated by phase interfaces [4], [7], [8]. Interface problems also exist in biological fluid dynamics which may involve the complex geometry and frequently the interaction of fluids with moving elastic structures. The study of blood flow in flexible tubes, cell dynamics and motility and the functioning of various physiological mechanisms requires solving interface problems [10], [11], [12], [13].

Mathematically, interface problems [14], [15], [16] usually lead to differential equations with discontinuous or non-smooth solutions across interfaces. Many numerical methods designed for smooth solutions do not work efficiently for interface problems. Interface problems of our interest may have one or more of the following properties:

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The coefficients of the differential equations may be discontinuous across the interfaces.
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The solutions and their derivatives may be discontinuous across the interfaces.
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The interfaces may be fixed or moving.
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There may be more than one interface.

The curved interface in interface problems causes several fundamental difficulties in the development and analysis of numerical schemes:

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Because of the low regularity of the solutions across the interface, most traditional numerical methods will not work efficiently.
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Generally interfaces can be arbitrary and complicated. Also analytical expressions for them are rarely available.
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Performing convergence analysis for interface problems in a conventional way is very difficult.

To solve the interface problems, traditional FEMs or discontinuous Galerkin finite element methods (DG-FEMs) require body-fitting meshes, i.e., it is aligned with interface. Otherwise, the convergence of method might be impaired, see [17], [18], [19]. However, when the interface has a complex geometry, generating a high-quality interface-fitted mesh is a time-consuming process. Also, this mesh generation process will be harder for problems with shape-changing or moving interface. In order to overcome these issues numerical methods based on non-body-fitting meshes have been introduced and studied. These methods are divided into the followings two classes based on the technique used near the interface to impose the jump conditions.

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The first approach is to modify shape functions around the interface according to the behaviors of solution, for instance, the extended FEM [20], [21], the multi-scale FEM [22], [23], and the partition of unity method [24], [25], also the Immersed FEM (IFEM).
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The second approach is to modify the variational formulation of the numerical scheme near the interface using suitable penalties, such as the Cut-FEM [26], [27], [28], [29]. Also, we refer readers to [1], [3], [30], [31], [32], [33], [34], [35], [36] for immersed interface method (IIM) and [37], [38], [38], [39], [40], [41] for matched interface and boundary (MIB) method.

A prominent feature of IFEM is that it uses structured or even Cartesian meshes to solve problems with nontrivial interface geometry because this method allows the interface to be immersed in some of the elements (See Fig. 1). In this method, elements are divided into two groups of interface elements (whose interior intersects with the interface) and regular elements (consisting of the rest of elements) (See Fig. 2). Standard shape functions are utilized on regular elements, while interface elements due to the non-smoothness solution at the interface need treatment that is done by modifying the basic functions. Li [42], for the first time, introduced a linear IFEM for the one-dimensional problems with one interface point. Later Adjerid and Lin [43], [44], [45], [46], [47], [48] developed the IFE functions with an arbitrary polynomial degree.

Recently by imitating IFEM, which combines the idea of immersed methods with the classical FEM, the immersed idea has been combined with non-classical FEM and finite volume method, such as immersed Petrov–Galerkin method [49], [50], nonconforming IFEM [51], [52], [53], immersed discontinuous Galerkin method [54], [55], partially penalized immersed finite element method (PP-IFEM) [47], [54], [56], [57], [58] and immersed finite volume method [59], [60], [61].

The weak Galerkin finite element method (WG-FEM), which was first planned and analyzed by Wang and Ye [62], raised as a new class of FEM for solving PDEs [63], [64]. In fact, the main idea of this method is to define generalized differential operators instead of classical differential operators involved in the variational form and to employ proper stabilizations to enforce weak continuities for approximating functions. In other words, the weak finite element function $u_{h} = {u_{0 h}, u_{b h}}$ is used in which $u_{0 h}$ is totally discontinuous and the component $u_{b h}$ on element boundary may be independent of the component $u_{0 h}$ in the interior of element. Thus the WG-FEM enforces the continuity weakly in the variational formulation. Furthermore, it is highly flexible in element construction and mesh generation. The WG-FEM has been developed successfully for a large range of mathematical and engineering problems (see e.g. [64], [65], [66], [67], [68], [69], [70], [71]).

Lately, the WG-FEM has been presented for elliptic interface problems [64], [72], [73], [74], [75]. This WG-FEM, like FEM, requires body-fitting meshes. L. Mu and X. Zhang proposed IWG-FEM by imitating IFEM for simulating elliptic interface problems [76]. The introduced method in [76] is the combination of IFE and WG methods. The IWG-FEM [76], [77] has the following advantages over standard WG-FEM [64], [72], [73], [74] and PP-IFEM [53], [56], respectively:

1.
It uses non-body fitting meshes such as structured or even Cartesian meshes.
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The matrix assembling process of IWG-FEM compared with PP-IFEM that needs information from neighboring elements is more efficient because all computations are done locally within an element.

The authors in [76] obtained the optimal error estimate of the IWG-FEM solutions in energy norm for the interface problem under a piecewise $H^{3}$ smoothness assumption on solution. Whereas, due to given $f \in L^{2} (Ω)$ , the solution to (2.1) only has the piecewise $H^{2}$ smoothness in general. This motivates us to investigate whether the IWG-FEM developed in [76] can converge optimally under the natural piecewise smoothness assumption instead of the artificial $H^{3}$ smoothness assumption on solution. For this purpose, we introduce a new energy norm that is stronger than the one used in [76] and we show the bilinear form in the IWG-FEM has both the continuity and coercivity in this energy norm. These properties along with the IFE interpolation error developed in [47] enable us to obtain an error bound for the IWG-FEM solution in the energy norm under the natural $H^{2}$ regularity assumption on solution. As an important result, the improved estimation further enables us to prove the optimal convergence in the $L^{2}$ norm for the IWG-FEM, which has not been established in [76].

The rest of this paper is organized as follows. In Section 2, we offer elliptic interface problem and define some notations. We provide an IWG-FE space based on WG and IFE spaces and supply an IWG-FE scheme for problem (2.1) in Section 3. Next, in Section 4 we recommend a new energy norm and discuss the well-posedness of the IWG-FE scheme in this energy norm. We show error estimate of IWG-FE scheme in the energy norm under the natural piecewise $H^{2}$ smoothness assumption in Section 5. Also we show the optimal convergence in the $L^{2}$ norm. Section 6 contains three numerical tests to illustrate the efficiency of suggested method and to confirm the theoretical rates of convergence. Finally, conclusions are drawn in Section 7.

Throughout the paper, the general positive constant $C$ in our analysis may be different at different situations.

Section snippets

Preliminaries

Consider a square domain $Ω \subset R^{2}$ that is separated by a smooth curve $Γ$ into two sub-domains $Ω^{+}$ and $Ω^{-}$ (See Fig. 3). We seek solutions of the $2 D$ elliptic equation with piecewise constant and discontinuous diffusion coefficient $α$ across the interface $Γ$ given by $\begin{matrix} - div (α \nabla u) = f, i n Ω^{+} \cup Ω^{-}, \\ u = g, o n \partial Ω, \end{matrix}$ with assumption that the solution and the normal component of the flux are continuous along $Γ$ , i.e., $〚 u 〛_{Γ} ≔ u^{-} |_{Γ} - u_{Γ}^{+} = 0,$ $〚 α \nabla u \cdot n 〛_{Γ} ≔ α^{-} \nabla u^{-} \cdot n |_{Γ} - α^{+} \nabla u^{+} \cdot n |_{Γ} = 0,$ where $n$ is the unit normal vector to the interface $Γ$ .

Immersed weak Galerkin finite element scheme

Let $T_{h} = {E}$ be a Cartesian triangular of $Ω$ (See Fig. 1). Let us define mesh diameter $h = max {h_{E}}$ , where $h_{E}$ is the diameter of element $E$ . For any element $E \in T_{h}$ , $E^{0}$ and $\partial E$ express the interior and the boundary of $E$ , respectively. Denote by $E_{h}$ the set of all edges in $T_{h}$ , and let $E_{h}^{0} = E_{h} ∖ \partial Ω$ be the set of all interior edges. A weak function on the element $E$ refers to $v = {v_{0}, v_{b}} \in L^{2} (E) \times H^{1 / 2} (\partial E)$ . The components $v_{0}$ and $v_{b}$ are the value of $v$ in the interior and on the boundary of $E$ , respectively. $W (E)$ denotes

Solvability

This section is devoted to investigate the existence and uniqueness of solution of IWG-FEM derived from (3.15). To this end, we introduce norms in the space $V_{h}$ as follows: ${‖ v ‖}_{h}^{2} = a (v_{0}, v_{0}) + S_{ρ} (v, v),$ and ${⦀ v ⦀}_{h}^{2} = {‖ v ‖}_{h}^{2} + ρ^{- 1} \sum_{E \in T_{h}} h_{E} {‖ Q_{b} (α \nabla v_{0} \cdot n) ‖}_{\partial E}^{2} .$ In fact, it is easy to see that ${‖ v ‖}_{h}$ and ${⦀ v ⦀}_{h}$ are norms on $V_{h}$ and ${‖ v ‖}_{h} \leq {⦀ v ⦀}_{h} .$ Now we recall the following useful lemmas.

Lemma 4.1

Trace Inequality for Regular Element [79]

Let $E \in T_{h}^{R}$ , for every function $v \in H^{1} (E)$ there exists a constant $C_{R}$ such that $h_{E} {‖ v ‖}_{\partial E}^{2} \leq C_{R} ({‖ v ‖}_{E}^{2} + h_{E}^{2} {‖ \nabla v ‖}_{E}^{2}),$ and if $v$ is a polynomial, using the

Error analysis

In this section, we derive optimal estimates for the errors of IWG-FE solution under the usual $H^{2}$ smoothness assumption on solution. We will first recall some results for elliptic problems.

Lemma 5.1

Let $E \in T_{h}$ , then for any $v \in P H^{2} (E)$ we have $‖ α \nabla v \cdot n - Q_{b} (α \nabla v \cdot n) ‖_{\partial E} \leq {‖ α \nabla (v - Q_{0} v) \cdot n ‖}_{\partial E} .$

Proof

By definition of $Q_{b}$ and adding zero in the form $\pm {〈 α \nabla v \cdot n - Q_{b} (α \nabla v \cdot n), (α \nabla Q_{0} v) \cdot n 〉}_{\partial E}$ for any $v \in P H^{2} (E)$ , we have ${‖ α \nabla v \cdot n - Q_{b} (α \nabla v \cdot n) ‖}_{\partial E}^{2} = {〈 α \nabla v \cdot n - Q_{b} (α \nabla v \cdot n), α \nabla v \cdot n - Q_{b} (α \nabla v \cdot n) 〉}_{\partial E} = {〈 α \nabla v \cdot n - Q_{b} (α \nabla v \cdot n), α \nabla v \cdot n - (α \nabla Q_{0} v) \cdot n 〉}_{\partial E} + 〈 α \nabla v \cdot n - Q_{b} (α \nabla v \cdot n), (α \nabla Q_{0} v) \cdot n - Q_{b} (α \nabla v \cdot n) 〉$

Numerical experiments

In this section, we will present some numerical tests to validate the accuracy and efficiency of the IWG-FEM for the elliptic interface problems. For all tests, the domain $Ω$ will be considered as $[- 1, 1] \times [- 1, 1]$ which is divided by interface $Γ$ into two sub-domains $Ω^{-} ≔ {(x, y) : Γ (x, y) < 0}$ and $Ω^{+} = Ω ∖ Ω^{-}$ . Also we take $ρ = 10 α_{m a x} ≔ 10 max {α^{+}, α^{-}}$ . The functions $f$ and $g$ in (2.1) are chosen according to the exact solution. We note that according to analysis done in the previous section, in Section 6.1 we consider

Conclusions

In this article, we employ a new analysis framework to derive the error bounds for the immersed weak Galerkin finite element method (IWG-FEM) developed in [76]. This new framework uses an energy norm ${⦀ \cdot ⦀}_{h}$ which is stronger than ${‖ \cdot ‖}_{h}$ norm originally used in [76]. There are two key-components in this analysis framework. First, it employs an optimal approximation capability for the gradient of the IFE interpolation with the natural smoothness assumption of solution on the boundary of interface

Acknowledgments

The authors are very grateful to two reviewers for carefully reading this paper and for their comments and suggestions, which have improved the paper.

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Citation Excerpt :
However, it is a non-trivial process to generate a high-quality interface-fitted mesh to solve complicated three-dimensional interface problems. On the other hand, the alternative unfitted mesh methods have been developed to solve interface problems such as the immersed interface FEMs [3,4], the immersed FEMs [5–8], the Cut FEM [9], extended FEMs [10,11], immersed weak Galerkin FEM [12–14] and so on. Meanwhile, many meshless methods have also been developed for the interface problems, such as, the meshless local Petrov-Galerkin method [15,16], the staggered meshless method [17], the meshless point collocation moving least squares (MLS) method [18], the interpolating stabilized MLS method (ISMLS) [19,20], the meshless collocation method [21–23], the generalized finite difference method (GFDM) [24], the meshless method based on Pascal polynomials and multiple-scale approach [25], and the meshless method using deep neural network [26,27].
This article presents a meshless method to solve three-dimensional elliptic interface problem. The method is based on the generalized finite difference method, which expresses the derivatives of unknown variables by linear combinations of nearby function values. The proposed method turns the interface problem into some boundary value sub-problems coupled by the interface conditions. This conversion leads to an important feature, that is, the proposed method is not sensitive to the jump coefficients or the interface's geometry. It can handle different complex interfaces by only changing the level set function of the interface in the process. Several numerical examples with sufficient complexity verify the accuracy and stability of the method. For some given examples, the method is more accurate than the classical immersed finite element method.

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