High-order finite difference schemes for elliptic problems with intersecting interfaces

Angelova, Ivanka Tr.; Vulkov, Lubin G.

doi:10.1016/j.amc.2006.08.165

Cited by 14 publications

(19 citation statements)

References 15 publications

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“…Thus, it is only necessary to take a few internal mesh points N and M for having good results, as it is shown in the numerical example. Comparing with results of [7] the local truncation error is of the same order in variable x and higher accurate in variable y. Because of qualitative properties of the numerical solution (136) with just a few nodes the quality of the approximation is better, saving computational cost.…”

Section: Discussionmentioning

confidence: 72%

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Closed form numerical solutions of variable coefficient linear second-order elliptic problems

Casabán¹,

Company²,

Jódar³

2014

Applied Mathematics and Computation

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In this work we develop an alternative numerical technique which allows to construct a numerical solution in closed form of variable coefficient linear second-order elliptic problems with Dirichlet boundary conditions. The elliptic partial differential equation is approximated by a consistent explicit difference scheme and using a discrete separation of the variables method we determine a closed form solution of the two resulting discrete boundary value problems with the separated variables, avoiding to have to solve large algebraic systems. One of these boundary value problems is a discrete SturmLiouville problem which guarantees the qualitative properties of the exact solution of elliptic problem. A constructive procedure for the computation of the numerical solution is given and an illustrative example is included.

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Section: Discussionmentioning

confidence: 72%

“…Apart from some techniques such as meshless methods [1,2] and those based on particular transformations used to solve special problems [3,4], the most used are related mesh methods as the finite difference method [5,6,7], the finite-volume method [8,9] and the finite element method [10,11].…”

Section: Introductionmentioning

confidence: 99%

Closed form numerical solutions of variable coefficient linear second-order elliptic problems

Casabán¹,

Company²,

Jódar³

2014

Applied Mathematics and Computation

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“…In this section we present numerical experiments obtained by applying the difference schemes (3), (4). The test problem is …”

Section: Numerical Experimentsmentioning

confidence: 99%

“…∞ is the global pointwise maximum norm and C is a constant independent of ε and N . In general, the gradients of the solution [2,3,13] become unbounded in the boundary/interface and corner layers as ε → 0; however, parameter-uniform numerical methods guarantee that the error in the numerical approximation is controlled solely by the size of N . Let us consider the boundary value problems for the reaction-diffusion model: where Ω = (−1, 1) × (0, 1), the diagonal matrix…”

Section: Introductionmentioning

confidence: 99%

Uniform Convergence of Finite-Difference Schemes for Reaction-Diffusion Interface Problems

Angelova

Vulkov

2008

Large-Scale Scientific Computing

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Abstract. We consider a singularly perturbed reaction-diffusion equation in two dimensions (x, y) with concentrated source on a segment parallel to axis Oy. By means of an appropriate (including corner layer functions) decomposition, we describe the asymptotic behavior of the solution. Finite difference schemes for this problem of second and fourth order of local approximation on Shishkin mesh are constructed. We prove that the first scheme is almost second order uniformly convergent in the maximal norm. Numerical experiments illustrate the theoretical order of convergence of the first scheme and almost fourth order of convergence of the second scheme.2000 Mathematics Subject Classification: 65M06, 65M12.

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“…Interface problems occur in many physical applications [1–4] such as composite materials [5, 6], fluid mechanics [7, 8], cell and bubble formation, crystal growth [9–11], biochemical processing, mining, material sciences [12], biological systems and sciences [13, 14], heat conduction [15], and heat and mass transfer [16–20].…”

Section: Introductionmentioning

confidence: 99%

Some numerical techniques for solving elliptic interface problems

Kumar

Joshi

2010

Numerical Methods Partial

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Many physical phenomena can be modeled by partial differential equations with singularities and interfaces. The standard finite difference and finite element methods may not be successful in giving satisfactory numerical results for such problems. Hence, many new methods have been developed. Some of them are developed with the modifications in the standard methods, so that they can deal with the discontinuities and the singularities. In this article, a survey has been done on some recent efficient techniques to solve elliptic interface problems.

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High-order finite difference schemes for elliptic problems with intersecting interfaces

Cited by 14 publications

References 15 publications

Closed form numerical solutions of variable coefficient linear second-order elliptic problems

Closed form numerical solutions of variable coefficient linear second-order elliptic problems

Uniform Convergence of Finite-Difference Schemes for Reaction-Diffusion Interface Problems

Some numerical techniques for solving elliptic interface problems

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