A high-order embedded domain method combining a Predictor–Corrector-Fourier-Continuation-Gram method with an integral Fourier pseudospectral collocation method for solving linear partial differential equations in complex domains

Abstract

Partial differential equations (PDEs) arise naturally in a wide variety of scientific areas and applications, and their numerical solutions are highly indispensable in many cases. Typical spectral/pseudospectral (PS) methods for solving PDEs work well only for regular domains such as rectangles or disks; however, the application of these methods to irregular domains is not straightforward and difficult enough to consider them less appealing as numerical tools. This research paper endeavors to take advantage of these methods in complex domains by introducing a novel, high-order numerical method that brings into play domain embedding into a regular, square computational domain in combination with integral reformulation, fully exponentially convergent Fourier PS collocation, and the Fourier-Continuation (FC)-Gram method integrated with a novel predictor–corrector-continuation algorithm to improve the accuracy of the extrapolated data. We developed some new formulas to construct the first- and second-order Fourier integration matrices (FIMs) based on equally-spaced nodes within the interval of integration. Modified FIMs were also developed to compute integral approximations of smooth, periodic functions when the upper limits of integration are any random points in the interval of integration. An algorithm for the fast and economic construction of the first-order FIM was also derived. A rounding error analysis based on numerical simulations demonstrates that the rounding errors in the calculation of the elements of the developed FIMs of size N are roughly of order less than or equal to O ( N u R ) , where u R ≈ 2 . 22 × 1 0 − 16 is the unit round-off of the double-precision floating-point system. The powerful features of the proposed method are illustrated through the study of the numerical solution of two-dimensional linear PDEs of Poisson type with constant coefficients and two different sets of boundary conditions. Two test problems are presented to demonstrate the overall capabilities of the proposed method. The current research paper investigations can be very useful when dealing with more complicated problems such as nonlinear PDEs in complex domains, or optimal control problems governed by linear/nonlinear PDEs in complex domains.