Direct and iterative solution of the generalized Dirichlet–Neumann map for elliptic PDEs on square domains

Abstract

MSC: 35J25 65N35 64N99 65F05 65F10 Keywords: Elliptic PDEs Dirichlet-Neumann map Global relation Collocation Iterative methods Jacobi Gauss-Seidel GMRES Bi-CGSTAB a b s t r a c tIn this work we derive the structural properties of the Collocation coefficient matrix associated with the Dirichlet-Neumann map for Laplace's equation on a square domain. The analysis is independent of the choice of basis functions and includes the case involving the same type of boundary conditions on all sides, as well as the case where different boundary conditions are used on each side of the square domain. Taking advantage of said properties, we present efficient implementations of direct factorization and iterative methods, including classical SOR-type and Krylov subspace (Bi-CGSTAB and GMRES) methods appropriately preconditioned, for both Sine and Chebyshev basis functions. Numerical experimentation, to verify our results, is also included.

Chapter 6: Overview of Variational Formulations for Linear Elliptic PDEs

“…• The only works on the numerical implementation of the Fokas method not included in Table 6.1 are the five papers [92], [91], [85], [52], and [7]. The first three [90] (they analyze the properties of the system of linear equations for specific geometries).…”

“…The papers [55], [90], [92], [91], [85], and [7] all effectively consider solving BVPs involving the operator ∂ /∂ z. The simplest such BVP is the following: let Ω be a bounded domain in with boundary Γ .…”

Chapter 1: Introduction

“…This rigorous approach is summarized in Chapter 4. Regarding numerical results, we note that the unified transform has inspired a novel numerical technique for the determination of the unknown boundary values [21], [22], [23], [24], [25], [26], [27], [28], [29], [30]. For elliptic PDEs formulated in the interior of a polygon, the above technique provides the analogue of the boundary integral method, but now the analysis takes place in the spectral (Fourier) space instead of the physical space.…”

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

“…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 [13][14][15][16][17][18][19][20][21]. Substantial progress in this direction was made by Fornberg and co-worker [14,15].…”