High-order numerical schemes based on difference potentials for 2D elliptic problems with material interfaces

Albright, James N.; Epshteyn, Yekaterina; Medvinsky, M.; Xia, Qing

doi:10.1016/j.apnum.2016.08.017

Cited by 19 publications

(37 citation statements)

References 42 publications

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“…Here, we define the fully-discrete finite-difference discretization of (3,4), and then define the Auxiliary Problem. Indeed, the discretization we consider is of time-and spatial-discretizations.…”

Section: Dpmmentioning

confidence: 99%

“…Proof. See [67] for the general theory of DPM (including the proof for general elliptic PDE), or one of [3,5,6] for the proof in the case of parabolic interface problems.…”

Section: Dpmmentioning

confidence: 99%

“…There is extensive existing work addressing numerical approximation of PDE posed on composite domains with interfaces or irregular domains, for example, the boundary integral method [11,56], difference potentials method [3,6,26,27,58,67], immersed boundary method [30,42,61,74], immersed interface method [2,46,47,49,69], ghost fluid method [31,32,50,51], the matched interface and boundary method [82,84,85,86], Cartesian grid embedded boundary method [19,41,57,83], multigrid method for elliptic problems with discontinuous coefficients on an arbitrary interface [18], virtual node method [9,39], Voronoi interface method [35,36], the finite difference method [8,10,24,75,78,79] and finite volume method [22,34] based on mapped grids, or cut finite element method [13,14,15,…”

Section: Introductionmentioning

confidence: 99%

“…For recent work on SBP-SAT-FD for wave equations in composite domains, see [10,17,75,78], and the two review papers [21,73]; for recent work in DPM for elliptic/parabolic problems in composite domains with interface defined explicitly, see [3,4,5,6,26,27,28,58,59,67]; and for recent work in cut-FEM see [13,14,15,37,53,71].…”

Section: Introductionmentioning

confidence: 99%

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High-Order Numerical Methods for 2D Parabolic Problems in Single and Composite Domains

et al. 2018

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In this work, we discuss and compare three methods for the numerical approximation of constant-and variable-coefficient diffusion equations in both single and composite domains with possible discontinuity in the solution/flux at interfaces, considering (i) the Cut Finite Element Method; (ii) the Difference Potentials Method; and (iii) the summationby-parts Finite Difference Method. First we give a brief introduction for each of the three methods. Next, we propose benchmark problems, and consider numerical tests-with respect to accuracy and convergence-for linear parabolic problems on a single domain, and continue with similar tests for linear parabolic problems on a composite domain (with the interface defined either explicitly or implicitly). Lastly, a comparative discussion of the methods and numerical results will be given.

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Section: Dpmmentioning

confidence: 99%

“…Proof. See [67] for the general theory of DPM (including the proof for general elliptic PDE), or one of [3,5,6] for the proof in the case of parabolic interface problems.…”

Section: Dpmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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High-Order Numerical Methods for 2D Parabolic Problems in Single and Composite Domains

et al. 2018

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“…In [54], the difference potential method with second-order accuracy in the solution and in the gradient has been developed for elliptic interface problems with variable coefficients in [15]. The fourth-order extension of the method for the elliptic interface problems is developed in [2]. …”

Section: Introductionmentioning

confidence: 99%

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

Li¹,

Ji²,

Chen³

2017

SIAM J. Numer. Anal.

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A new augmented method is proposed for elliptic interface problems with a piecewise variable coefficient that has a finite jump across a smooth interface. The main motivation is not only to get a second order accurate solution but also a second order accurate gradient from each side of the interface. The key of the new method is to introduce the jump in the normal derivative of the solution as an augmented variable and re-write the interface problem as a new PDE that consists of a leading Laplacian operator plus lower order derivative terms near the interface. In this way, the leading second order derivatives jump relations are independent of the jump in the coefficient that appears only in the lower order terms after the scaling. An upwind type discretization is used for the finite difference discretization at the irregular grid points near or on the interface so that the resulting coefficient matrix is an M-matrix. A multi-grid solver is used to solve the linear system of equations and the GMRES iterative method is used to solve the augmented variable. Second order convergence for the solution and the gradient from each side of the interface has also been proved in this paper. Numerical examples for general elliptic interface problems have confirmed the theoretical analysis and efficiency of the new method.

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High‐order numerical method for 2D biharmonic interface problem

Tameh

Shakeri

2022

Numerical Methods in Fluids

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We present a robust and effective method for the numerical solution of the biharmonic interface problem with discontinuities in both the solution and its derivatives. We use a mixed scheme, in which the biharmonic equation is decoupled to two Poisson equations. The proposed approach is based on the method of difference potentials combined with finite difference schemes on regular structured grid to solve this problem with high‐order accuracy on nonconforming domains. Representative numerical experiments confirm the accuracy and effectiveness of the proposed method and its ability to handle problems with coupled equations.

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High-order numerical schemes based on difference potentials for 2D elliptic problems with material interfaces

Cited by 19 publications

References 42 publications

High-Order Numerical Methods for 2D Parabolic Problems in Single and Composite Domains

High-Order Numerical Methods for 2D Parabolic Problems in Single and Composite Domains

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

High‐order numerical method for 2D biharmonic interface problem

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