Numerical Solution for Elliptic Interface Problems Using Spectral Element Collocation Method

Hessari, Peyman; Kim, Sang Dong; Shin, Bo Sung

doi:10.1155/2014/780769

Cited by 8 publications

(5 citation statements)

References 15 publications

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“…If T is one-to-one, it is invertible [27]. By the implicit function theorem [27], if the Jacobian of the transformation T ,ŷ)) defined on the rectangular domain S, and the elliptic equation defined on curved domain Ω is also transformed to an elliptic equation defined on the rectangular domain S. For more details and examples, see [12,15].…”

Section: Discussionmentioning

confidence: 99%

Pseudospectral Least Squares Method for Stokes--Darcy Equations

Hessari¹

2015

SIAM J. Numer. Anal.

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Abstract. We investigate the first order system least squares Legendre and Chebyshev pseudospectral method for coupled Stokes-Darcy equations. A least squares functional is defined by summing up the weighted L 2 -norm of residuals of the first order system for coupled Stokes-Darcy equations and that of Beavers-Joseph-Saffman interface conditions. Continuous and discrete homogeneous functionals are shown to be equivalent to a combination of weighted H(div) and H 1 -norms for Stokes and Darcy equations. The spectral convergence for the Legendre and Chebyshev methods is derived. Some numerical experiments are demonstrated to validate our analysis.

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Section: Discussionmentioning

confidence: 99%

Pseudospectral Least Squares Method for Stokes--Darcy Equations

Hessari¹

2015

SIAM J. Numer. Anal.

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“…Eq. (19), assumes the form (21) u, ξξ + D (22) u, ξη + D (23) u,ηη + D (11) u, ξ + D (12) u,η = f Similarly the eigenvalue problem, corresponding to Eq. (20), can be written.…”

Section: {︃mentioning

confidence: 99%

“…The latter takes advantage of the blending function interpolation used to transform the quadrilateral regions [21]. So far, this technique has been successfully associated with some pseudospectral methods [22,23].…”

Section: Introductionmentioning

confidence: 99%

Solving biharmonic problems on irregular domains by stably evaluated Gaussian kernel

Krowiak

2020

Curved and Layered Structures

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The paper extends recently developed idea of stable evaluation of the Gaussian kernel. Owing to this, the Gaussian radial basis function that is sensitive to the shape parameter can be stably evaluated and applied to interpolation problems as well as to solve differential equations, giving highly accurate results. But it can be done only with grids being the Cartesian product of sets of points, what limits the use of this idea to rectangular domains. In the present paper, by the association of an appropriate transformation with the mentioned method, the latter is applied to solve biharmonic problems on quadrilateral irregular domains. As an example, in the present work this approach is applied to solve bending as well as free vibration problem of thin plates. In the paper some strategies for the implementation of the boundary conditions are also presented and examined due to the accuracy. The numerical tests show high accuracy and usefulness of the method.

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“…Using the norm equivalence for the homogeneous functional

G_{w} (\cdot; 0)

, we used the following block diagonal matrix as a preconditioner of

A_{N}

to apply PCGM to solve pseudospectral collocation approximation of the problem :

{scriptP}_{N} = true(\begin{matrix} ν^{2} W_{N} \\ ν^{2} S_{N} \\ ν^{2} S_{N} \\ W_{N} \end{matrix} true),

where

W_{N}

is the diagonal weight matrix and

S_{N}

is the finite element stiffness matrix for the \hbox{Poisson} problem based on the piecewise continuous bilinear functions with Gauss–Lobatto points as the nodal points. The implementation of spectral collocation method with more details can be found in .…”

Section: Implementation and Numerical Testsmentioning

confidence: 99%

“…Remark For the Stokes equations defined in any arbitrary domain Ω , application of spectral element method along with the so‐called Gordon‐Hall map is needed. For details of Gordon‐Hall map and numerous examples see .…”

Section: Implementation and Numerical Testsmentioning

confidence: 99%

Least‐squares spectral method for velocity‐vorticity‐pressure form of the Stokes equations

Shin

Hessari

2015

Numerical Methods Partial

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The aim of this article is to present and analyze first-order system least-squares spectral method for the Stokes equations in two-dimensional spaces. The Stokes equations are transformed into a first-order system of equations by introducing vorticity as a new variable. The least-squares functional is then defined by summing up the L 2 w -and H −1 w -norms of the residual equations. The H −1 w -norm in the least-squares functional is replaced by suitable operator. Continuous and discrete homogeneous least-squares functionals are shown to be equivalent to H 1 w -norm of velocity and L 2 w -norm of vorticity and pressure for spectral Galerkin and pseudospectral method. The spectral convergence of the proposed methods are given and the theory is validated by numerical experiment. Mass conservation is also briefly investigated.

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Numerical Solution for Elliptic Interface Problems Using Spectral Element Collocation Method

Cited by 8 publications

References 15 publications

Pseudospectral Least Squares Method for Stokes--Darcy Equations

Pseudospectral Least Squares Method for Stokes--Darcy Equations

Solving biharmonic problems on irregular domains by stably evaluated Gaussian kernel

Least‐squares spectral method for velocity‐vorticity‐pressure form of the Stokes equations

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