8 research outputs found
Supercloseness and asymptotic analysis of the Crouzeix-Raviart and enriched Crouzeix-Raviart elements for the Stokes problem
For the Crouzeix-Raviart and enriched Crouzeix-Raviart elements, asymptotic
expansions of eigenvalues of the Stokes operator are derived by establishing
two pseudostress interpolations, which admit a full one-order supercloseness
with respect to the numerical velocity and the pressure, respectively. The
design of these interpolations overcomes the difficulty caused by the lack of
supercloseness of the canonical interpolations for the two nonconforming
elements, and leads to an intrinsic and concise asymptotic analysis of
numerical eigenvalues, which proves an optimal superconvergence of eigenvalues
by the extrapolation algorithm. Meanwhile, an optimal superconvergence of
postprocessed approximations for the Stokes equation is proved by use of this
supercloseness. Finally, numerical experiments are tested to verify the
theoretical results
The Lower Bounds for Eigenvalues of Elliptic Operators --By Nonconforming Finite Element Methods
The aim of the paper is to introduce a new systematic method that can produce
lower bounds for eigenvalues. The main idea is to use nonconforming finite
element methods. The general conclusion herein is that if local approximation
properties of nonconforming finite element spaces are better than global
continuity properties of , corresponding methods will produce lower bounds
for eigenvalues. More precisely, under three conditions on continuity and
approximation properties of nonconforming finite element spaces we first show
abstract error estimates of approximate eigenvalues and eigenfunctions.
Subsequently, we propose one more condition and prove that it is sufficient to
guarantee nonconforming finite element methods to produce lower bounds for
eigenvalues of symmetric elliptic operators. As one application, we show that
this condition hold for most nonconforming elements in literature. As another
important application, this condition provides a guidance to modify known
nonconforming elements in literature and to propose new nonconforming elements.
In fact, we enrich locally the Crouzeix-Raviart element such that the new
element satisfies the condition; we propose a new nonconforming element for
second order elliptic operators and prove that it will yield lower bounds for
eigenvalues. Finally, we prove the saturation condition for most nonconforming
elements.Comment: 24 page
Computing the lower and upper bounds of Laplace eigenvalue problem: by combining conforming and nonconforming finite element methods
This article is devoted to computing the lower and upper bounds of the
Laplace eigenvalue problem. By using the special nonconforming finite elements,
i.e., enriched Crouzeix-Raviart element and extension , we get
the lower bound of the eigenvalue. Additionally, we also use conforming finite
elements to do the postprocessing to get the upper bound of the eigenvalue. The
postprocessing method need only to solve the corresponding source problems and
a small eigenvalue problem if higher order postprocessing method is
implemented. Thus, we can obtain the lower and upper bounds of the eigenvalues
simultaneously by solving eigenvalue problem only once. Some numerical results
are also presented to validate our theoretical analysis.Comment: 19 pages, 4 figure
Lower bounds of eigenvalues of the biharmonic operators by the rectangular Morley element methods
In this paper, we analyze the lower bound property of the discrete
eigenvalues by the rectangular Morley elements of the biharmonic operators in
both two and three dimensions. The analysis relies on an identity for the
errors of eigenvalues. We explore a refined property of the canonical
interpolation operators and use it to analyze the key term in this identity. In
particular, we show that such a term is of higher order for two dimensions, and
is negative and of second order for three dimensions, which causes a main
difficulty. To overcome it, we propose a novel decomposition of the first term
in the aforementioned identity. Finally, we establish a saturation condition to
show that the discrete eigenvalues are smaller than the exact ones. We present
some numerical results to demonstrate the theoretical results
A Multi-level Correction Scheme for Eigenvalue Problems
In this paper, a new type of multi-level correction scheme is proposed for
solving eigenvalue problems by finite element method. With this new scheme, the
accuracy of eigenpair approximations can be improved after each correction step
which only needs to solve a source problem on finer finite element space and an
eigenvalue problem on the coarsest finite element space. This correction scheme
can improve the efficiency of solving eigenvalue problems by finite element
method.Comment: 16 pages, 5 figure