It is well known that the approximation of eigenvalues and associated eigenfunctions of a linear operator under constraint is a difficult problem. One of the difficulties is to propose methods of approximation which satisfy in a stable and accurate way the eigenvalues equations, the constraint one and the boundary conditions. Using any non-stable method leads to the presence of non-physical eigenvalues: a multiple zero one called spurious modes and non-zero one called pollution modes. One way to eliminate these two families is to favor the constraint equations by satisfying it exactly and to verify the equations of the eigenvalues equations in weak ways. To illustrate our contribution in this field we consider in this paper the case of Stokes operator. We describe several methods that produce the correct number of eigenvalues. We numerically prove how these methods are adequate to correctly solve the 2D Stokes eigenvalue problem.

We investigate the cubature points based triangular spectral element method and provide accuracy results for elliptic problems in non polygonal domains using various isoparametric mappings. The capabilities of the method are here again clearly confirmed.

In this paper we perform and analyze a Karhunen-Loève expansion on the solution of a discrete heat equation. Unlike the continuous case, several choices can be made from the numerical scheme to the numerical time integration. We analyze some of these choices and compare them. In literature, it is shown that the KL-expansion’s error depends on the singular values of the matrix (or the operator) which we attempt to compress, but there is few results on the decay of these singular values. The core of this article is to prove the exponential decay of the singular values. To achieve this result, the analysis is conducted in a classical way by considering the spatial correlation of the temperature. And then, we analyze the problem in a more general view by using the Krylov matrices with Hermitian argument, which are a generalization of the well-known Vandermonde matrices. Some computations are made using MATLAB to ensure the performance of the Karhunen-Loève expansion. This article presents an application of this work to an identification problem where the data are disturbed by a Gaussian white noise.

We develop a domain decomposition Chebyshev spectral collocation method for the second-kind linear and nonlinear Volterra integral equations with smooth kernel functions. The method is easy to implement and possesses high accuracy. In the convergence analysis, we derive the spectral convergence order under the $L^∞$-norm without the Chebyshev weight function, and we also show numerical examples which coincide with the theoretical analysis.

In this paper, we propose numerical schemes for stochastic differential equations driven by white noise and colored noise, respectively. For this purpose, we first discretize the white noise and colored noise, and give their regularity estimates. Then we use spectral element methods to solve the corresponding stochastic differential equations numerically. The approximation errors are derived, and the numerical results demonstrate high accuracy of the proposed schemes.

Efficient and unconditionally stable high order time marching schemes are very important but not easy to construct for nonlinear phase dynamics. In this paper, we propose and analysis an efficient stabilized linear Crank-Nicolson scheme for the Cahn-Hilliard equation with provable unconditional stability. In this scheme the nonlinear bulk force are treated explicitly with two second-order linear stabilization terms. The semi-discretized equation is a linear elliptic system with constant coefficients, thus robust and efficient solution procedures are guaranteed. Rigorous error analysis show that, when the time step-size is small enough, the scheme is second order accurate in time with a prefactor controlled by some lower degree polynomial of $1/ \varepsilon.$ Here $\varepsilon$ is the interface thickness parameter. Numerical results are presented to verify the accuracy and efficiency of the scheme.