A comparison of the finite difference and multiresolution method for the elliptic equations with Dirichlet boundary conditions on irregular domains

https://doi.org/10.1016/j.jcp.2021.110207Get rights and content

Highlights

  • A multiresolution method for elliptic equations on irregular domain is presented.

  • The multiresolution method yields up to eighth-order for error and gradient.

  • The multiresolution method preconditions the resulted linear system.

  • We compare the second-order finite difference method on uniform and adaptive grids.

Abstract

We make a comparison of the finite difference and multiresolution method for solving the elliptic equations on irregular domains. The Dirichlet boundary condition is treated by the ghost fluid method (GFM) for the finite difference method and the Lagrange multiplier for the multiresolution method. Numerical results illustrate the improved convergence rate of errors and their gradients with the multiresolution method, up to eighth-order. Moreover, the wavelet-based multiresolution has an advantage of preconditioning of the linear system arising from the discretization of elliptic equation over the finite difference method.

Introduction

Elliptic equations are the fundamental equations of PDEs (Partial differential equations), which are mainly used to describe the equilibrium and stable state of physics, such as the electromagnetic field, gravitational field and reaction diffusion phenomenon under steady state. Various methods have been developed to solve the equations on irregular domains. The finite difference method is the most commonly used numerical method due to its easy and flexible implementation on the computer [1], [2]. In [3], a finite difference method was used to discretize the variable coefficient Poisson equation with Dirichlet boundary condition on an irregular interface with uniform Cartesian grid. This method yields a symmetric discretization matrix and second-order-accurate solution. For the interface point, a linear extrapolation of the GFM was applied to impose the boundary condition to ensure the symmetry of the produced matrix. However, it was noted in [4] that both second-order truncation errors at the boundary and internal points are not adequate to produce comparable error with the second-order discretization on regular domains since the boundary error dominates. To obtain much higher accurate solution and its gradient, the quadratic extrapolation of the GFM which results in third order accuracy at the boundary, is recommended although the symmetry of matrix is lost. Moreover, [5] analyzed the influence of different choices of ghost value and interface location on the accuracy of solution and its gradient. It concluded that the quadratic extrapolation and quadratic interpolation of finding interface location result in solution and its gradient of second-order accuracy. A fourth-order accuracy discretization on Laplace equations imposed on irregular domains was developed in [6]. The authors utilized a fourth order discretization coupled with the GFM for the interface treatment in a manner similar to the approach in [3]. A more general robin boundary condition [7] was considered to solve the Poisson equation, the heat equation and Stefan problem on irregular domains. It used the finite volume method [8] to impose the boundary condition on an uniform grid with nodes centered at the cells. This method yields a symmetric linear system and turns to be of second-order accuracy for the Poisson equation, the heat equation and first-order accuracy for the Stefan-type problem. A recent work in [9] used GFM for the case of Robin boundary condition with piecewise smooth surface. It presented a new integration procedure on piecewise smooth surface in the second-order accurate finite volume method. The numerical examples showed second-order accurate solution and gradients in two and three spatial dimensions.

In order to improve the computation efficiency and save CPU memory, we only need to put fine resolution on a small portion of the domain near the interface. Therefore, the adaptive mesh technique is required. The authors in [10], [11] proposed to use non-graded Cartesian grids to solve the variable coefficient Poisson equation and got second-order accuracy of both solutions and their gradients. The main advantages of this approach are that the grids is automatically produced based on its proximity to the irregular interface and represented by the quadtree (in 2D) and octree (in 3D) data structure. In addition, a brief comparison with the second-order finite difference method with linear extrapolation on uniform grids [3] which yields first-order accuracy of gradient, was conducted. It showed the discretization on adaptive grids is more efficient in terms of number of nodes and computational time. The finite difference scheme on non-graded adaptive Cartesian grids was applied to solve the Stefan problem which involves solving the heat equations with Dirichlet boundary condition on each phase and locating the moving interface [12]. It demonstrates supralinear convergence rate for some physical effects such as surface tension and crystalline anisotropy. In the case of the diffusion and Stefan problems with Robin boundary conditions, the authors used adaptive Cartesian grids coupled with a hybrid finite difference method for space discretization and finite volume method for Robin boundary conditions treatment [13]. This approach is proven to be of second-order accuracy for the diffusion equation and first-order accuracy for the Stefan problem in two or three space dimension. An improvement was made in [14] by preserving the third-order accuracy in implementing finite volume method for the robin boundary conditions on uniform grids. It used a second-order five-point difference scheme for internal grid points and achieved second-order solutions and their gradients. A review of using the finite difference method to solve the elliptic and parabolic problems on uniform and adaptive grids is done in [15]. It detailed three important methods, i.e., level set method for capturing irregular domain's interface, the GFM for imposing Dirichlet boundary condition, and an ungraded adaptive grid for accelerating computation efficiency. The main advantage of this approach is that it is applicable to a wide variety of equations with complex interfaces while yielding second-order solutions. In the case where the jump condition is enforced on irregular domain, some variants of the GFM are considered. The GFM was shown to produce non-convergent gradients on uniform grid due to the smearing of the tangential component of the jump. Applying the GFM on a Voronoi grid was proven to be of second-order accuracy of the solution and first-order accuracy of its gradient [16]. The authors in [17] proposed another way of recovering convergence of gradient by an iterative process to enforce the jump condition while preserving the symmetric positive-definiteness structure of the standard Poisson discretization. In addition, it is pointed in [18] that choosing ghost values in the fast-diffusion region produces linear system with small condition numbers regardless of the ratio of the diffusion coefficients.

Multiresolution method is a powerful mathematical tool. It has important applications in signal and image processing, time series analysis and approximation theory. And the wavelet transform based multiresolution method has received increasing interests in solving elliptic equations due to its preconditioning of arising linear system and adaptive approximation of high accuracy [19]. The author used the fictitious domain method combined with Lagrange multiplier approach to solve the elliptic equations on irregular domains with mixed boundary condition [20]. Then the Galerkin-wavelet method was employed to discretize its resulted weak form. The advantage of using Lagrange multiplier approach is that it separates the treatment of differential operator from the boundary condition. This is particularly important in the numerical simulation of PDEs with moving boundary. A Petrov-Galerkin wavelet method is proposed to solve the saddle point problem while yielding an identity stiff matrix [21]. The author proved the existence, uniqueness and convergence of numerical solution and wavelet preconditioning of the linear system. However, the Lagrange multiplier used to enforce the boundary condition is interpreted as the conormal derivative of numerical solution on the boundary. The convergence rate of the method presented above is no more than 32. A variant of the fictitious domain method called a smooth fictitious domain method was proposed in [22]. It puts the Lagrange multipliers on an auxiliary boundary located outside the original domain and results the convergence rate to around 2 by the finite element discretization. We combined the smooth fictitious domain method and multiresolution method to solve elliptic equation on irregular domain and obtain a much higher convergence rate related to the multiresolution order (up to 6 order). Moreover the enhanced convergence rate is justified by a proved interior error estimate in 1D and 2D [23], [24].

In this work, we compare the finite difference method with the multiresolution method on solving the elliptic equation on irregular domains in the respects of error and its gradient, convergence rate and condition number of arising linear systems. This paper is organized as follows. Section 2 presents the formulation of the finite difference method (on uniform and adaptive grids) and the multiresolution method for solving the equation on irregular domain. It details the treatment of boundary condition and gradient computation. Section 3 gives two examples on circle and rotated-leaf domain to illustrate the error and convergence rate of each method. We conclude in Section 4.

Section snippets

The equation and numerical methods

We shall consider the elliptic equation with the Dirichlet boundary condition,{(Iν)u=f in ω,u=g on γ, where ω is an irregular domain bounded by γ, △ is Laplace operator, functions f, g and constant ν>0 are given. In this paper, the level set method is used to implicitly represent the interface. This approach uses a signed distance function ϕ, i.e.,{ϕ<0 for xω,ϕ>0 for xω,ϕ=0 on xγ. The main advantage of this approach is that it can detect and deal with topological change at the interface

Examples

In each example, we consider the equation (Iν)u=f with Dirichlet boundary condition. The original domain ω is embedded in a fictitious domain Ω=[0,1]×[0,1].

Conclusion

We have analyzed the finite difference method (on uniform and adaptive grid) and the multiresolution method for solving the elliptic equation on irregular domain with Dirichlet boundary condition. While the second-order finite difference scheme with linear extrapolation for defining the ghost values and the fourth-order finite difference scheme with cubic extrapolation on uniform grid yields the solution in second and fourth-order respectively, a higher order of extrapolation (quadratic and

CRediT authorship contribution statement

Ping Yin: Conceptualization, Methodology, Software. Jacques Liandrat: Resources, Supervision. Wanqiang Shen: Visualization.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The research of P. Yin was funded by the National Natural Science Foundation of China (No. 11801222) and the Natural Science Foundation of Jiangsu Province of China for Young Scholar (No. Bk20150124). The research of W.Q. Shen was funded by the National Natural Science Foundation of China (No. 61772013).

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