A numerical technique for linear elliptic partial differential equations in polygonal domains

Abstract

Integral representations for the solution of linear elliptic partial differential equations (PDEs) can be obtained using Green's theorem. However, these representations involve both the Dirichlet and the Neumann values on the boundary, and for a well-posed boundary-value problem (BVPs) one of these functions is unknown. A new transform method for solving BVPs for linear and integrable nonlinear PDEs usually referred to as the unified transform (or the Fokas transform) was introduced by the second author in the late Nineties. For linear elliptic PDEs, this method can be considered as the analogue of Green's function approach but now it is formulated in the complex Fourier plane instead of the physical plane. It employs two global relations also formulated in the Fourier plane which couple the Dirichlet and the Neumann boundary values. These relations can be used to characterize the unknown boundary values in terms of the given boundary data, yielding an elegant approach for determining the Dirichlet to Neumann map. The numerical implementation of the unified transform can be considered as the counterpart in the Fourier plane of the well-known boundary integral method which is formulated in the physical plane. For this implementation, one must choose (i) a suitable basis for expanding the unknown functions and (ii) an appropriate set of complex values, which we refer to as collocation points, at which to evaluate the global relations. Here, by employing a variety of examples we present simple guidelines of how the above choices can be made. Furthermore, we provide concrete rules for choosing the collocation points so that the condition number of the matrix of the associated linear system remains low.

Linear Elliptic PDEs in a Cylindrical Domain with a Polygonal Cross‐Section

“…Furthermore, these authors have employed the so-called Halton nodes for collocation points and have utilized the crucial observation that the conditioning of the associated linear system improves if the linear system is overdetermined. The authors of [24], following Fornberg and coworker, have also used Legendre polynomials and have also overdetermined the relevant system; however, instead of using Halton nodes, they have used the collocation points suggested in [23]. In this way, the authors of [24] have been able to improve substantially the relevant condition number.…”

“…The authors of [24], following Fornberg and coworker, have also used Legendre polynomials and have also overdetermined the relevant system; however, instead of using Halton nodes, they have used the collocation points suggested in [23]. In this way, the authors of [24] have been able to improve substantially the relevant condition number. The main advantage of the method developed in [24] compared with standard numerical methods available for solving linear elliptic PDEs (including finite-difference methods, finite element methods, boundary integral equations, and the method of particular solutions) follows from the fact that it involves a boundary-based discretization that does not require the computation of singular integrals (as opposed to the discretizations of the usual boundary integral equations).…”

“…For the corresponding two-dimensional problem [24], and for the particular case that a Dirichlet boundary condition is prescribed on every side, it is rigorously shown in [8,9] [25] and [26] that for the Dirichlet problem, the global relations determine uniquely the unknown Neumann boundary values.…”

“…We define the functionsP,Q,Ř by (24)- (26) with α = 1 and with λ 1 , λ ∈ C satisfying the constraint (31). Then, (28) is a divergence form for the Laplace equation.…”

“…(vi) The analysis of the global relations yields a novel numerical technique for the numerical solution of the generalized Dirichlet-to-Neumann map, i.e., for the determination of the unknown boundary values in terms of the prescribed boundary data [15][16][17][18][19][20][21][22][23]. In particular, substantial progress in this direction was made by Fornberg and coauthors [16,17], as well as in [24]. This paper is organized as follows.…”