A decomposed immersed interface method for variable coefficient elliptic equations with non-smooth and discontinuous solutions

Berthelsen, Petter Andreas

doi:10.1016/j.jcp.2003.12.003

Cited by 95 publications

(92 citation statements)

References 19 publications

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“…We demonstrate the accuracy of our method when applied to problems considered by previous authors [4,13,14,28]. For interfaces with moderate curvature, it is never significantly worse and sometimes better.…”

Section: Comparison and Accuracy Studymentioning

confidence: 68%

“…A fixed Cartesian grid, where the interface can cut through the grid lines, is often used. A variety of methods have been proposed to deal with the grid-interface interaction [4,7,9,10,16,[13][14][15]28,32,[29][30][31]33,34].…”

Section: Introductionmentioning

confidence: 99%

“…Another second-order method for elliptic interface problems is the decomposed immersed interface method (DIIM), which decomposes the jump data along coordinate directions [4]. The method uses the standard central finite difference scheme for the left-hand side and introduces a correction term from jumps to the righthand side, where high-order one-sided interpolation is used on both sides of the interface.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Piecewise-polynomial discretization and Krylov-accelerated multigrid for elliptic interface problems

Chen

Strain

2008

Journal of Computational Physics

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A new piecewise-polynomial interface method (PIM) for discretizing elliptic problems with complex interfaces between high-contrast materials is derived, analyzed and tested. A Krylov-accelerated interface multigrid approach (IMG) solves the discretization efficiently. Stability and convergence are proved in one dimension, while an extensive array of numerical experiments with complex interfaces and large coefficient transitions demonstrate the accuracy, efficiency and robustness of the method in two dimensions.

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Section: Comparison and Accuracy Studymentioning

confidence: 68%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Piecewise-polynomial discretization and Krylov-accelerated multigrid for elliptic interface problems

Chen

Strain

2008

Journal of Computational Physics

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“…Since the pioneer work of Peskin [50] in 1977, much attention has been paid to the numerical solution of elliptic equations with discontinuous coefficients and singular sources on regular Cartesian grids [7,8,11,15,18,20,[30][31][32]55,58]. Simple Cartesian grids are preferred in these studies since the complicated procedure of generating unstructured grid could be bypassed, and well developed fast algebraic solvers could be utilized.…”

Section: Introductionmentioning

confidence: 99%

“…A relevant, while quite distinct approach is the integral equation method for complex geometry [44,45]. Aforementioned methods have found much success in scientific and engineering applications [6][7][8]15,18,20,[25][26][27][28]30,32,34,39,41,40,42,53,54,[57][58][59]. A possible further direction in the field could be the development of higher order interface methods [20,60,61] which are particularly desirable for problems involving both material interfaces and high frequency oscillations, such as the interaction of turbulence and shock, and high frequency wave propagation in inhomogeneous media [5].…”

Section: Introductionmentioning

confidence: 99%

Matched interface and boundary (MIB) method for elliptic problems with sharp-edged interfaces

Zhou

Guo

2007

Journal of Computational Physics

169

182

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Elliptic problems with sharp-edged interfaces, thin-layered interfaces and interfaces that intersect with geometric boundary, are notoriously challenging to existing numerical methods, particularly when the solution is highly oscillatory. This work generalizes the matched interface and boundary (MIB) method previously designed for solving elliptic problems with curved interfaces to the aforementioned problems. We classify these problems into five distinct topological relations involving the interfaces and the Cartesian mesh lines. Flexible strategies are developed to systematically extends the computational domains near the interface so that the standard central finite difference scheme can be applied without the loss of accuracy. Fictitious values on the extended domains are determined by enforcing the physical jump conditions on the interface according to the local topology of the irregular point. The concepts of primary and secondary fictitious values are introduced to deal with sharp-edged interfaces. For corner singularity or tip singularity, an appropriate polynomial is multiplied to the solution to remove the singularity. Extensive numerical experiments confirm the designed second order convergence of the proposed method.

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Computational methods for optical molecular imaging

Chen

Wei

Cong

et al. 2008

Commun. Numer. Meth. Engng.

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SummaryA new computational technique, the matched interface and boundary (MIB) method, is presented to model the photon propagation in biological tissue for the optical molecular imaging. Optical properties have significant differences in different organs of small animals, resulting in discontinuous coefficients in the diffusion equation model. Complex organ shape of small animal induces singularities of the geometric model as well. The MIB method is designed as a dimension splitting approach to decompose a multidimensional interface problem into one-dimensional ones. The methodology simplifies the topological relation near an interface and is able to handle discontinuous coefficients and complex interfaces with geometric singularities. In the present MIB method, both the interface jump condition and the photon flux jump conditions are rigorously enforced at the interface location by using only the lowest-order jump conditions. This solution near the interface is smoothly extended across the interface so that central finite difference schemes can be employed without the loss of accuracy. A wide range of numerical experiments are carried out to validate the proposed MIB method. The second-order convergence is maintained in all benchmark problems. The fourth-order convergence is also demonstrated for some three-dimensional problems. The robustness of the proposed method over the variable strength of the linear term of the diffusion equation is also examined. The performance of the present approach is compared with that of the standard finite element method. The numerical study indicates that the proposed method is a potentially efficient and robust approach for the optical molecular imaging.

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A decomposed immersed interface method for variable coefficient elliptic equations with non-smooth and discontinuous solutions

Cited by 95 publications

References 19 publications

Piecewise-polynomial discretization and Krylov-accelerated multigrid for elliptic interface problems

Piecewise-polynomial discretization and Krylov-accelerated multigrid for elliptic interface problems

Matched interface and boundary (MIB) method for elliptic problems with sharp-edged interfaces

Computational methods for optical molecular imaging

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