Spectral Method and Spectral Element Method for Three Dimensional Linear Elliptic System: Analysis and Application In this paper, a leastsquares spectral method and a nonconforming leastsquares spectral element method for three dimensional linear elliptic system will be presented. Differentiability estimates and the main stability theorem for the proposed method are proven. Using the regularity estimate and the proposed stability estimates, we introduce a suitable preconditioner and show that the condition number of the preconditioned system is (O((ln W)2)) (where W is degree of polynomials). Then we establish the error estimates of the proposed method. Specific numerical results based on linear elasticity problem, fourth order problem and sixth order problem are discussed to reflect the efficiency of the proposed method.

Highorder Gegenbauer integral spectral element method integrated with an adaptive Chebyshev optimization strategy for solving linear singularly perturbed differential equations Abstract   We propose a highorder adaptive spectral element method based on Gegenbauer polynomial basis for solving singularly perturbed differential equations with Dirichlet boundary conditions and a boundary layer at one end. The method reduces the singularly perturbed boundary value problem into a single system of linear algebraic equations when the solution is smooth on the solution domain, or two systems of linear algebraic equations when the solution has a thin boundary layer. The core of the new adaptive strategy lies in constructing a mathematical model of the last few Gegenbauer coefficients in the Gegenbauer truncated series approximating the secondorder derivative of the solution, which can be readily converted into a minimization problem in a leastsquares sense and solved using fast optimization techniques. An error and convergence analysis of the method for sufficiently smooth solutions is presented. A numerical test example with various parameter settings is presented to verify the accuracy, effectiveness, and applicability of the proposed method. Numerical comparisons with other rival methods in the literature are included to demonstrate further the power of the proposed method.