Accurate Solution and Gradient Computation for Elliptic Interface Problems with Variable Coefficients

Li, Zhilin; Ji, Haifeng; Chen, Xiaohong

doi:10.1137/15m1040244

Cited by 39 publications

(10 citation statements)

References 63 publications

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“…The simple problem can be solved by many different numerical methods, such as immersed boundary method (IBM) [27], immersed interface method (IIM) [18], and variants of finite element methods [5,6,13,15]. It is known that IBM for the problem only yields first-order accuracy in the maximum norm [21,26] and IIM generates second-order accuracy in the maximum norm for both the numerical solution and its gradient [2,22,23].…”

Section: Introductionmentioning

confidence: 99%

Sharp error estimates of a fourth‐order compact scheme for a Poisson interface problem

Dong

Ying

Zhang

2020

Numerical Methods Partial

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A simple and efficient fourth-order kernel-free boundary integral method was recently proposed by Xie and Ying for constant coefficients elliptic PDEs on complex domains. This method is constructed by a compact finite difference scheme and works efficiently with fourth-order accuracy in the maximum norm. But it is challenging to present the sharp error analysis of the resulting approach since the local truncation errors, at the irregular grid nodes near the interface, are only in the order of O(h 3). The aim of this paper is to establish rigorous sharp error analysis. We prove that both the numerical solution and its gradient have fourth-order accuracy in the discrete 2-norm, and the scheme has fourth-order accuracy in the maximum norm based on the properties of discrete Green functions. Numerical examples are also provided to verify the error analysis.

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Section: Introductionmentioning

confidence: 99%

Sharp error estimates of a fourth‐order compact scheme for a Poisson interface problem

Dong

Ying

Zhang

2020

Numerical Methods Partial

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“…The main idea of the IIM is to include the jump conditions in the discretization by utilizing a multivariate Taylor expansion. This was later extended in [2,14,15], thereby obtaining a second-order accurate solution and gradient at the interface. Recently, IIM with global second-order convergence in gradients was developed in [31].…”

Section: Introductionmentioning

confidence: 99%

A second-order accurate semi-Lagrangian method for convection-diffusion equations with interfacial jumps

Cho¹,

Park²,

Kang³

2020

Preprint

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In this paper, we present a second-order accurate finite-difference method for solving convectiondiffusion equations with interfacial jumps on a moving interface. The proposed method is constructed under a semi-Lagrangian framework for convection-diffusion equations; a novel interpolation scheme is developed in the presence of jump conditions. Combined with a second-order ghost fluid method [3], a sharp capturing method with a first-order local truncation error near the interface and second-order truncation error away from the interface is developed for the convectiondiffusion equation. In addition, a level-set advection algorithm is presented when the velocity gradient jumps across the interface. Numerical experiments support the conclusion that the proposed methods for convection-diffusion equations and level-set advection are necessary for the second-order convergence solution and the interface position.

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“…However, the global second-order convergence of the gradient of the solution obtained by IIM is still unclear. For recent related work, see [1,11].…”

Section: Introductionmentioning

confidence: 99%

A Second-Order Boundary Condition Capturing Method for Solving the Elliptic Interface Problems on Irregular Domains

et al. 2019

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A second-order boundary condition capturing method is presented for the elliptic interface problem with jump conditions in the solution and its normal derivative. The proposed method is an extension of the work in Liu et al. (J Comput Phys 160(1):151-178, 2000) to a higher order. The motivation of proposed method is that the approximated value at the interface can be reconstructed by proper interpolation based on the level set representation from Gibou et al. (J Comput Phys 176(1):205-227, 2002). A second-order accurate method is constructed, both in the solution and its gradient, using second-order finite difference approximation. Several numerical results demonstrate that the proposed method is indeed second-order accurate in the solution and its gradient in the L 2 and L ∞ norms.

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Accurate Solution and Gradient Computation for Elliptic Interface Problems with Variable Coefficients

Cited by 39 publications

References 63 publications

Sharp error estimates of a fourth‐order compact scheme for a Poisson interface problem

Sharp error estimates of a fourth‐order compact scheme for a Poisson interface problem

A second-order accurate semi-Lagrangian method for convection-diffusion equations with interfacial jumps

A Second-Order Boundary Condition Capturing Method for Solving the Elliptic Interface Problems on Irregular Domains

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