41 research outputs found
On the mesh nonsingularity of the moving mesh PDE method
The moving mesh PDE (MMPDE) method for variational mesh generation and
adaptation is studied theoretically at the discrete level, in particular the
nonsingularity of the obtained meshes. Meshing functionals are discretized
geometrically and the MMPDE is formulated as a modified gradient system of the
corresponding discrete functionals for the location of mesh vertices. It is
shown that if the meshing functional satisfies a coercivity condition, then the
mesh of the semi-discrete MMPDE is nonsingular for all time if it is
nonsingular initially. Moreover, the altitudes and volumes of its elements are
bounded below by positive numbers depending only on the number of elements, the
metric tensor, and the initial mesh. Furthermore, the value of the discrete
meshing functional is convergent as time increases, which can be used as a
stopping criterion in computation. Finally, the mesh trajectory has limiting
meshes which are critical points of the discrete functional. The convergence of
the mesh trajectory can be guaranteed when a stronger condition is placed on
the meshing functional. Two meshing functionals based on alignment and
equidistribution are known to satisfy the coercivity condition. The results
also hold for fully discrete systems of the MMPDE provided that the time step
is sufficiently small and a numerical scheme preserving the property of
monotonically decreasing energy is used for the temporal discretization of the
semi-discrete MMPDE. Numerical examples are presented.Comment: Revised and improved version of the WIAS preprin
How a nonconvergent recovered Hessian works in mesh adaptation
Hessian recovery has been commonly used in mesh adaptation for obtaining the
required magnitude and direction information of the solution error.
Unfortunately, a recovered Hessian from a linear finite element approximation
is nonconvergent in general as the mesh is refined. It has been observed
numerically that adaptive meshes based on such a nonconvergent recovered
Hessian can nevertheless lead to an optimal error in the finite element
approximation. This also explains why Hessian recovery is still widely used
despite its nonconvergence. In this paper we develop an error bound for the
linear finite element solution of a general boundary value problem under a mild
assumption on the closeness of the recovered Hessian to the exact one.
Numerical results show that this closeness assumption is satisfied by the
recovered Hessian obtained with commonly used Hessian recovery methods.
Moreover, it is shown that the finite element error changes gradually with the
closeness of the recovered Hessian. This provides an explanation on how a
nonconvergent recovered Hessian works in mesh adaptation.Comment: Revised (improved proofs and a better example
On the mesh nonsingularity of the moving mesh PDE method
The moving mesh PDE (MMPDE) method for variational mesh generation and adaptation is studied theoretically at the discrete level, in particular the nonsingularity of the obtained meshes. Meshing functionals are discretized geometrically and the MMPDE is formulated as a modified gradient system of the corresponding discrete functionals for the location of mesh vertices. It is shown that if the meshing functional satisfies a coercivity condition, then the mesh of the semi-discrete MMPDE is nonsingular for all time if it is nonsingular initially. Moreover, the altitudes and volumes of its elements are bounded below by positive numbers depending only on the number of elements, the metric tensor, and the initial mesh. Furthermore, the value of the discrete meshing functional is convergent as time increases, which can be used as a stopping criterion in computation. Finally, the mesh trajectory has limiting meshes which are critical points of the discrete functional. The convergence of the mesh trajectory can be guaranteed when a stronger condition is placed on the meshing functional. Two meshing functionals based on alignment and equidistribution are known to satisfy the coercivity condition. The results also hold for fully discrete systems of the MMPDE provided that the time step is sufficiently small and a numerical scheme preserving the property of monotonically decreasing energy is used for the temporal discretization of the semi-discrete MMPDE. Numerical examples are presente
Stability of explicit one-step methods for P1-finite element approximation of linear diffusion equations on anisotropic meshes
We study the stability of explicit one-step integration schemes for the
linear finite element approximation of linear parabolic equations. The derived
bound on the largest permissible time step is tight for any mesh and any
diffusion matrix within a factor of , where is the spatial
dimension. Both full mass matrix and mass lumping are considered. The bound
reveals that the stability condition is affected by two factors. The first one
depends on the number of mesh elements and corresponds to the classic bound for
the Laplace operator on a uniform mesh. The other factor reflects the effects
of the interplay of the mesh geometry and the diffusion matrix. It is shown
that it is not the mesh geometry itself but the mesh geometry in relation to
the diffusion matrix that is crucial to the stability of explicit methods. When
the mesh is uniform in the metric specified by the inverse of the diffusion
matrix, the stability condition is comparable to the situation with the Laplace
operator on a uniform mesh. Numerical results are presented to verify the
theoretical findings.Comment: Revised WIAS Preprin
A comparative numerical study of meshing functionals for variational mesh adaptation
We present a comparative numerical study for three functionals used for
variational mesh adaptation. One of them is a generalisation of Winslow's
variable diffusion functional while the others are based on equidistribution
and alignment. These functionals are known to have nice theoretical properties
and work well for most mesh adaptation problems either as a stand-alone
variational method or combined within the moving mesh framework. Their
performance is investigated numerically in terms of equidistribution and
alignment mesh quality measures. Numerical results in 2D and 3D are presented.Comment: Additional example (H1), journal referenc
A geometric discretization and a simple implementation for variational mesh generation and adaptation
We present a simple direct discretization for functionals used in the
variational mesh generation and adaptation. Meshing functionals are discretized
on simplicial meshes and the Jacobian matrix of the continuous coordinate
transformation is approximated by the Jacobian matrices of affine mappings
between elements. The advantage of this direct geometric discretization is that
it preserves the basic geometric structure of the continuous functional, which
is useful in preventing strong decoupling or loss of integral constraints
satisfied by the functional. Moreover, the discretized functional is a function
of the coordinates of mesh vertices and its derivatives have a simple
analytical form, which allows a simple implementation of variational mesh
generation and adaptation on computer. Since the variational mesh adaptation is
the base for a number of adaptive moving mesh and mesh smoothing methods, the
result in this work can be used to develop simple implementations of those
methods. Numerical examples are given.Comment: Corrected and improved versio
Recommended from our members
A study on the conditioning of finite element equations with arbitrary anisotropic meshes via a density funtion approach
The linear finite element approximation of a general linear diffusion problem with arbitrary anisotropic meshes is considered. The conditioning of
the resultant stiffness matrix and the Jacobi preconditioned stiffness matrix
is investigated using a density function approach proposed by Fried in 1973.
It is shown that the approach can be made mathematically rigorous for general
domains and used to develop bounds on the smallest eigenvalue and the
condition number that are sharper than existing estimates in one and two
dimensions and comparable in three and higher dimensions. The new results
reveal that the mesh concentration near the boundary has less influence on
the condition number than the mesh concentration in the interior of the
domain. This is especially true for the Jacobi preconditioned system where
the former has little or almost no influence on the condition number.
Numerical examples are presented