An analytical method for linear elliptic PDEs and its numerical implementation

Fulton, Scott R.; Fokas, A. S.; Xenophontos, Christos

doi:10.1016/j.cam.2003.10.012

Cited by 45 publications

(65 citation statements)

References 8 publications

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“…The implementation of the new method to the case of the Laplace equation in an arbitrary bounded convex polygon was presented in [11], where: where n is the number of the sides of the polygon, {z i } n i=1 are the corners of the polygon in the complex z-plane (with z n+1 = z 1 ) and N, which is chosen to be even, is the number of points used for the discrete approximation of the unknown boundary values. This choice has been motivated by the analytical integral representation in [6] (see also Remark 2.1).…”

Section: Introductionmentioning

confidence: 99%

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The generalized Dirichlet–Neumann map for linear elliptic PDEs and its numerical implementation

Sifalakis

Fokas

Fulton

et al. 2008

Journal of Computational and Applied Mathematics

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A new approach for analyzing boundary value problems for linear and for integrable nonlinear PDEs was introduced in Fokas [A unified transform method for solving linear and certain nonlinear PDEs, Proc. Roy. Soc. London Ser. A 53 (1997) 1411-1443]. For linear elliptic PDEs, an important aspect of this approach is the characterization of a generalized Dirichlet to Neumann map: given the derivative of the solution along a direction of an arbitrary angle to the boundary, the derivative of the solution perpendicularly to this direction is computed without solving on the interior of the domain. This is based on the analysis of the so-called global relation, an equation which couples known and unknown components of the derivative on the boundary and which is valid for all values of a complex parameter k. A collocation-type numerical method for solving the global relation for the Laplace equation in an arbitrary bounded convex polygon was introduced in Fulton et al. [An analytical method for linear elliptic PDEs and its numerical implementation, J. Comput. Appl. Math. 167 (2004) 465-483]. Here, by choosing a different set of the "collocation points" (values for k), we present a significant improvement of the results in Fulton et al. [An analytical method for linear elliptic PDEs and its numerical implementation, J. Comput. Appl. Math. 167 (2004) 465-483].The new collocation points lead to wellconditioned collocation methods. Their combination with sine basis functions leads to a collocation matrix whose diagonal blocks are point diagonal matrices yielding efficient implementation of iterative methods; numerical experimentation suggests quadratic convergence. The choice of Chebyshev basis functions leads to higher order convergence, which for regular polygons appear to be exponential.

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Section: Introductionmentioning

confidence: 99%

“…In this paper, aiming to improve and stabilize the order of convergence and the associated conditioning number of the collocation method in [11], we use a different set of collocation points (the values of the complex parameter k): we use the values specified by Eq. for the real part of the global relation.…”

Section: Introductionmentioning

confidence: 99%

The generalized Dirichlet–Neumann map for linear elliptic PDEs and its numerical implementation

Sifalakis

Fokas

Fulton

et al. 2008

Journal of Computational and Applied Mathematics

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“…Thus, low values for N often suffice for high accuracy. Most previous numerical implementations of this new analytical formulation have been limited to a relatively low O(N −2 ) algebraic rate of convergence (Fulton et al 2004;Fokas 2008;Fokas et al 2009), thus using relatively large N -values and iterative solution procedures. The possibility of spectral convergence (by discretizing with Chebyshev rather than Fourier expansions along the boundaries) was noted in Sifalakis et al (2008) and Smitheman et al (2010).…”

Section: Introductionmentioning

confidence: 99%

A numerical implementation of Fokas boundary integral approach: Laplace's equation on a polygonal domain

Fornberg

Flyer

2011

Proc. R. Soc. A.

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“…This property allows for efficient numerical integration, with the use of contour deformation to ensure exponential convergence of the integrands. Previous numerical studies have followed a pointwise evaluation of the global relation at special collocation points [8,9,13,14]. More recent work has resulted in similarly low condition numbers, and which are independent of β [7,10].…”

Section: Resultsmentioning

confidence: 95%

“…There have been several numerical implementations of this approach for a variety of boundary value problems [7,8,9,13,15,14]. The numerical experimentation in these works suggested that Fokas' unified method gives rise to an accurate and efficient means of numerically approximating the solution to elliptic boundary value problems.…”

Section: Introductionmentioning

confidence: 99%