102 research outputs found
Visualization of pentatopic meshes
We propose a simple tool to visualize 4D unstructured
pentatopic meshes. The method slices unstructured 4D
pentatopic meshes (fields) with an arbitrary 3D hyperplane and
obtains a conformal 3D unstructured tetrahedral representation
of the mesh (field) slice ready to explore with standard 3D
visualization tools. The results show that the method is suitable
to visually explore 4D unstructured meshes. This capability has
facilitated devising our 4D bisection method, and thus, we think
it might be useful when devising new 4D meshing methods.
Furthermore, it allows visualizing 4D scalar fields, which is a
crucial feature for our space-time application
Convergence and optimality of the adaptive Morley element method
This paper is devoted to the convergence and optimality analysis of the
adaptive Morley element method for the fourth order elliptic problem. A new
technique is developed to establish a quasi-orthogonality which is crucial for
the convergence analysis of the adaptive nonconforming method. By introducing a
new parameter-dependent error estimator and further establishing a discrete
reliability property, sharp convergence and optimality estimates are then fully
proved for the fourth order elliptic problem
Adaptive vertex-centered finite volume methods for general second-order linear elliptic PDEs
We prove optimal convergence rates for the discretization of a general
second-order linear elliptic PDE with an adaptive vertex-centered finite volume
scheme. While our prior work Erath and Praetorius [SIAM J. Numer. Anal., 54
(2016), pp. 2228--2255] was restricted to symmetric problems, the present
analysis also covers non-symmetric problems and hence the important case of
present convection
Adaptive Spectral Galerkin Methods with Dynamic Marking
The convergence and optimality theory of adaptive Galerkin methods is almost
exclusively based on the D\"orfler marking. This entails a fixed parameter and
leads to a contraction constant bounded below away from zero. For spectral
Galerkin methods this is a severe limitation which affects performance. We
present a dynamic marking strategy that allows for a super-linear relation
between consecutive discretization errors, and show exponential convergence
with linear computational complexity whenever the solution belongs to a Gevrey
approximation class.Comment: 20 page
Convergence and Optimality of Adaptive Mixed Finite Element Methods
The convergence and optimality of adaptive mixed finite element methods for
the Poisson equation are established in this paper. The main difficulty for
mixed finite element methods is the lack of minimization principle and thus the
failure of orthogonality. A quasi-orthogonality property is proved using the
fact that the error is orthogonal to the divergence free subspace, while the
part of the error that is not divergence free can be bounded by the data
oscillation using a discrete stability result. This discrete stability result
is also used to get a localized discrete upper bound which is crucial for the
proof of the optimality of the adaptive approximation
Strict bounding of quantities of interest in computations based on domain decomposition
This paper deals with bounding the error on the estimation of quantities of
interest obtained by finite element and domain decomposition methods. The
proposed bounds are written in order to separate the two errors involved in the
resolution of reference and adjoint problems : on the one hand the
discretization error due to the finite element method and on the other hand the
algebraic error due to the use of the iterative solver. Beside practical
considerations on the parallel computation of the bounds, it is shown that the
interface conformity can be slightly relaxed so that local enrichment or
refinement are possible in the subdomains bearing singularities or quantities
of interest which simplifies the improvement of the estimation. Academic
assessments are given on 2D static linear mechanic problems.Comment: Computer Methods in Applied Mechanics and Engineering, Elsevier,
2015, online previe
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