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

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 $2(d+1)$, where $d$ 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

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

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

Measurements and Code Comparison of Wave Dispersion and Antenna Radiation Resisitance for Helicon Waves in a High Density Cylindrical Plasma Source

Helicon wave dispersion and radiation resistance measurements in a high density (ne ≈ 1019 - 1020m-3) and magnetic field (B < 0.2 T) cylindrical plasma source are compared to the results of a recently developed numerical plasma wave code [I. V. Kamensk

Tetrahedral mesh improvement using moving mesh smoothing, lazy searching flips, and RBF surface reconstruction

Given a tetrahedral mesh and objective functionals measuring the mesh quality which take into account the shape, size, and orientation of the mesh elements, our aim is to improve the mesh quality as much as possible. In this paper, we combine the moving mesh smoothing, based on the integration of an ordinary differential equation coming from a given functional, with the lazy flip technique, a reversible edge removal algorithm to modify the mesh connectivity. Moreover, we utilize radial basis function (RBF) surface reconstruction to improve tetrahedral meshes with curved boundary surfaces. Numerical tests show that the combination of these techniques into a mesh improvement framework achieves results which are comparable and even better than the previously reported ones.Comment: Revised and improved versio

Structure and mechanism of the RNA polymerase II CTD phosphatase Scp1 and large-scale preparation of the RNA polymerase II-TFIIF complex

TFIIF is the only general transcription factor that has been implicated in the preinitiation complex assembly, open complex formation, initiation and transcription elongation. In addition, TFIIF stimulates Fcp1, a central phosphatase needed for recycling of RNA polymerase II (Pol II) after transcription by dephosphorylation of the Pol II C-terminal domain (CTD). This thesis reports the X-ray structure of the small CTD phosphatase Scp1, which is homologous to the Fcp1 catalytic domain. The structure shows a core fold and an active center similar to phosphotransferases and –hydrolases that solely share a DXDX(V/T) signature motif with Fcp1/Scp1. It was further demonstrated that the first aspartate in the signature motif undergoes metalassisted phosphorylation during catalysis, resulting in a phosphoaspartate intermediate that was structurally mimicked with the inhibitor beryllofluoride. Specificity may result from CTD binding to a conserved hydrophobic pocket between the active site and an insertion domain that is unique to Fcp1/Scp1. Fcp1 specificity may additionally arise from phosphatase recruitment near the CTD via the Pol II subcomplex Rpb4/7, which is shown to be required for Fcp1 binding to the polymerase in vitro. Until now, the main impediment in the high resolution crystallographic studies of TFIIF in complex with Pol II and other members of transcription machinery was unavailability of soluble, stoichiometric TFIIF complex in sufficient amounts. This thesis reports on the development of the overexpression system in yeast and a purification protocol that enabled for the first time to isolate milligram amounts of a pure and soluble, 15-subunit (~0,7 MDa) stoichiometric Pol IITFIIF complex. Such complex together with the promoter DNA, RNA, TBP and TFIIB assembles in vitro into the yeast initially transcribing complex, which can now be studied structurally