Efficient numerical solution of the generalized Dirichlet–Neumann map for linear elliptic PDEs in regular polygon domains

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.…”

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

“…Employing the expressions (19) in the constraint (8) yields the following constraint between λ 1 and λ:…”

“…(vi) The first steps have been taken toward extending the unified transform to three dimensions, see, for example [11,14]. (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].…”