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
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
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.
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