226 research outputs found
The MINI mixed finite element for the Stokes problem: An experimental investigation
Super-convergence of order 1.5 in pressure and velocity has been
experimentally investigated for the two-dimensional Stokes problem discretised
with the MINI mixed finite element. Even though the classic mixed finite
element theory for the MINI element guarantees linear convergence for the total
error, recent theoretical results indicate that super-convergence of order 1.5
in pressure and of the linear part of the computed velocity to the piecewise
linear nodal interpolation of the exact velocity is in fact possible with
structured, three-directional triangular meshes. The numerical experiments
presented here suggest a more general validity of super-convergence of order
1.5, possibly to automatically generated and unstructured triangulations. In
addition, the approximating properties of the complete computed velocity have
been compared with the approximating properties of the piecewise-linear part of
the computed velocity, finding that the former is generally closer to the exact
velocity, whereas the latter conserves mass better
A stabilized finite element method for the two-field and three-field Stokes eigenvalue problems
In this paper, the stabilized finite element approximation of the Stokes eigenvalue problems is considered for both the two-field (displacement–pressure) and the three-field (stress–displacement–pressure) formulations. The method presented is based on a subgrid scale concept, and depends on the approximation of the unresolvable scales of the continuous solution. In general, subgrid scale techniques consist in the addition of a residual based term to the basic Galerkin formulation. The application of a standard residual based stabilization method to a linear eigenvalue problem leads to a quadratic eigenvalue problem in discrete form which is physically inconvenient. As a distinguished feature of the present study, we take the space of the unresolved subscales orthogonal to the finite element space, which promises a remedy to the above mentioned complication. In essence, we put forward that only if the orthogonal projection is used, the residual is simplified and the use of term by term stabilization is allowed. Thus, we do not need to put the whole residual in the formulation, and the linear eigenproblem form is recovered properly. We prove that the method applied is convergent, and present the error estimates for the eigenvalues and the eigenfunctions. We report several numerical tests in order to illustrate that the theoretical results are validated.Peer ReviewedPostprint (author's final draft
A stabilized finite element method for the two-field and three-field Stokes eigenvalue problems
In this paper, the stabilized finite element approximation of the Stokes
eigenvalue problems is considered for both the two-field
(displacement-pressure) and the three-field (stress-displacement-pressure)
formulations. The method presented is based on a subgrid scale concept, and
depends on the approximation of the unresolvable scales of the continuous
solution. In general, subgrid scale techniques consist in the addition of a
residual based term to the basic Galerkin formulation. The application of a
standard residual based stabilization method to a linear eigenvalue problem
leads to a quadratic eigenvalue problem in discrete form which is physically
inconvenient. As a distinguished feature of the present study, we take the
space of the unresolved subscales orthogonal to the finite element space, which
promises a remedy to the above mentioned complication. In essence, we put
forward that only if the orthogonal projection is used, the residual is
simplified and the use of term by term stabilization is allowed. Thus, we do
not need to put the whole residual in the formulation, and the linear
eigenproblem form is recovered properly. We prove that the method applied is
convergent, and present the error estimates for the eigenvalues and the
eigenfunctions. We report several numerical tests in order to illustrate that
the theoretical results are validated
Finite element differential forms on curvilinear cubic meshes and their approximation properties
We study the approximation properties of a wide class of finite element
differential forms on curvilinear cubic meshes in n dimensions. Specifically,
we consider meshes in which each element is the image of a cubical reference
element under a diffeomorphism, and finite element spaces in which the shape
functions and degrees of freedom are obtained from the reference element by
pullback of differential forms. In the case where the diffeomorphisms from the
reference element are all affine, i.e., mesh consists of parallelotopes, it is
standard that the rate of convergence in L2 exceeds by one the degree of the
largest full polynomial space contained in the reference space of shape
functions. When the diffeomorphism is multilinear, the rate of convergence for
the same space of reference shape function may degrade severely, the more so
when the form degree is larger. The main result of the paper gives a sufficient
condition on the reference shape functions to obtain a given rate of
convergence.Comment: 17 pages, 1 figure; v2: changes in response to referee reports; v3:
minor additional changes, this version accepted for Numerische Mathematik;
v3: very minor updates, this version corresponds to the final published
versio
Convergence analysis of the scaled boundary finite element method for the Laplace equation
The scaled boundary finite element method (SBFEM) is a relatively recent
boundary element method that allows the approximation of solutions to PDEs
without the need of a fundamental solution. A theoretical framework for the
convergence analysis of SBFEM is proposed here. This is achieved by defining a
space of semi-discrete functions and constructing an interpolation operator
onto this space. We prove error estimates for this interpolation operator and
show that optimal convergence to the solution can be obtained in SBFEM. These
theoretical results are backed by a numerical example.Comment: 15 pages, 3 figure
On the existence and the uniqueness of the solution to a fluid-structure interaction problem
In this paper we consider the linearized version of a system of partial
differential equations arising from a fluid-structure interaction model. We
prove the existence and the uniqueness of the solution under natural regularity
assumptions
Least-squares formulations for eigenvalue problems associated with linear elasticity
We study the approximation of the spectrum of least-squares operators arising from linear elasticity. We consider a two-field (stress/displacement) and a three-field (stress/displacement/vorticity) formulation; other formulations might be analyzed with similar techniques. We prove a priori estimates and we confirm the theoretical results with simple two-dimensional numerical experiments.</p
- …