A direct method for accurate solution and gradient computations for elliptic interface problems

Chen, Xiaohong; Feng, Xiufang; Li, Zhilin

doi:10.1007/s11075-018-0503-5

Cited by 13 publications

(17 citation statements)

References 14 publications

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“…So that the accuracy can be guaranteed in the whole domain. The most significant advantage of the IIM is that it can achieve global second order accuracy [19,34]. Then the IIM was applied to Stokes and Navier-Stokes flows in [17,22].…”

Section: Introductionmentioning

confidence: 99%

A Simple 3D Immersed Interface Method for Stokes Flow with Singular Forces on Staggered Grids

Tan¹

2022

CiCP

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In this paper, a fairly simple 3D immersed interface method based on the CG-Uzawa type method and the level set representation of the interface is employed for solving three-dimensional Stokes flow with singular forces along the interface. The method is to apply the Taylor's expansions only along the normal direction and incorporate the jump conditions up to the second normal derivatives into the finite difference schemes. A second order geometric iteration algorithm is employed for computing orthogonal projections on the surface with third-order accuracy. The Stokes equations are discretized involving the correction terms on staggered grids and then solved by the conjugate gradient Uzawa type method. The major advantages of the present method are the special simplicity, the ability in handling the Dirichlet boundary conditions, and no need of the pressure boundary condition. The method can also preserve the volume conservation and the discrete divergence free condition very well. The numerical results show that the proposed method is second order accurate and efficient.

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Section: Introductionmentioning

confidence: 99%

A Simple 3D Immersed Interface Method for Stokes Flow with Singular Forces on Staggered Grids

Tan¹

2022

CiCP

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“…The immersed interface method (IIM) [7], which was introduced by Leveque and Li, is a widely known method for solving (1). It provides second-order accurate numerical solutions in the L ∞ norm.…”

Section: Introductionmentioning

confidence: 99%

“…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%

“…Extrapolation methods [2,13] are an alternative approach to solving (1). The boundary condition capturing method [13], which is also known as ghost fluid method (GFM), is a well-known example.…”

Section: Introductionmentioning

confidence: 99%

“…Another method for solving (1) is the correction function method proposed by Marques et al [14], where (1) is reformulated using correction functions. This enhances the accuracy of the numerical solution and its gradient up to fourth order in the L 2 norm.…”

Section: Introductionmentioning

confidence: 99%

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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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A high order compact finite difference scheme for elliptic interface problems with discontinuous and high-contrast coefficients

Feng

Han

Minev

2022

Applied Mathematics and Computation

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A direct method for accurate solution and gradient computations for elliptic interface problems

Cited by 13 publications

References 14 publications

A Simple 3D Immersed Interface Method for Stokes Flow with Singular Forces on Staggered Grids

A Simple 3D Immersed Interface Method for Stokes Flow with Singular Forces on Staggered Grids

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

A high order compact finite difference scheme for elliptic interface problems with discontinuous and high-contrast coefficients

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