Spectral and Pseudospectral Methods for Solving Linear Partial Differential Equations in Complex Domains: High-Order Hybrid Integral Fourier Pseudospectral Collocation, Fourier-Continuation-Gram, and Fourier Extensions of Arbitrary Length Methods

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 project endeavors to take advantage of these methods in complex domains by introducing some novel, high-order numerical methods that bring into play domain embedding into a regular, rectangular computational domain in combination with integral reformulation, fully exponentially convergent Fourier PS collocation, and Fourier extensions of arbitrary length (FE-AL) methods. The main features of the proposed methods are: (i) the ability to ensure exceedingly accurate solution approximations using relatively small collocation points, stable Fourier quadratures based on sparse and well-conditioned Fourier operational matrices of integration, Fourier PS collocation, Fourier-Continuation-Gram (FC-Gram) method, and FE-AL methods, (ii) the exponential convergence accomplished in spatial and temporal directions via the extension of the coefficients and the source term of the PDE in a smooth way to the embedding domain, (iii) the ease of accommodating irregular geometrical shapes, (iv) the simplicity in handling various types of boundary conditions, (v) sharing the same order of the running time as that of the usual spectral/PS collocation methods for PDEs on regular geometry, (vi) the significant reduction of the condition number of the resulting linear algebraic systems, (vii) dispensing with preconditioners typically used with standard spectral/PS methods and FC-based solvers, and (viii) avoid the degradation of accuracy due to the clustering of the collocation nodes near the boundary of the fictitious rectangle-- that is outside the irregular domain of interest-- typically associated with the use of collocation nodes of Gauss type. We shall focus our study on the numerical solution of two-dimensional linear PDEs and time-dependent one-dimensional linear PDEs of elliptic, parabolic, and hyperbolic types with constant, one-dimensional, and two-dimensional variable coefficients. The overall capabilities of the proposed methods will be demonstrated by means of testing the results against exact solutions and competitive numerical methods in the literature, in addition to applications to challenging real-life problems. We expect to establish some novel numerical methods that outperform conventional methods with respect to the condition number of the system matrix, accuracy, convergence rate, and efficiency. The current project investigations can be very useful when dealing with more complicated problems such as optimal control problems governed by linear PDEs in complex domains.