On the Fokas method for the solution of elliptic problems in both convex and non-convex polygonal domains

There exists a growing literature on using the Fokas method (unified transform method) to solve Laplace and Helmholtz problems on convex polygonal domains. We show here that the convexity requirement can be eliminated by the use of a ‘virtual side’ concept, thereby significantly increasing the flexibility and utility of the approach. We also show that the inclusion of singular functions in the basis to treat corner singularities can greatly increase the rate of convergence of the method. The method also compares well with other standard methods used to cope with corner singularities. An example is given where this inclusion leads to exponential convergence. As well as this, we give new results on several additional issues, including the choice of collocation points and calculation of solutions throughout domain interiors. An appendix illustrates the algebraic simplicity of the methodology by showing how the core part of the present approach can be implemented in only about a dozen lines of MATLAB code. •‘Virtual sides’ allow the study of non-convex domains and interface problems.•‘Singularity functions’ greatly improve accuracy and rate of convergence. This is also compared against standard methods in the literature.•Calculations of solutions throughout domain interiors are discussed.•Choice of collocation points and implementation parameters are discussed in detail.•Highly simplified Matlab code for the algorithm is given.