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Mesh sensitivity of finite element solution for linear elliptic partial differential equations is analyzed. A bound for the change in the finite element solution is obtained in terms of the mesh deformation and its gradient. The bound shows how the finite element solution changes continuously with the mesh. The result holds in any dimension and for arbitrary unstructured simplicial meshes, general linear elliptic partial differential equations, and general finite element approximations., 13 pages

A high-order quasi-conservative discontinuous Galerkin (DG) method is proposed for the numerical simulation of compressible multi-component flows. A distinct feature of the method is a predictor-corrector strategy to define the grid velocity. A Lagrangian mesh is first computed based on the flow velocity and then used as an initial mesh in a moving mesh method (the moving mesh partial differential equation or MMPDE method ) to improve its quality. The fluid dynamic equations are discretized in the direct arbitrary Lagrangian-Eulerian framework using DG elements and the non-oscillatory kinetic flux while the species equation is discretized using a quasi-conservative DG scheme to avoid numerical oscillations near material interfaces. A selection of one- and two-dimensional examples are presented to verify the convergence order and the constant-pressure-velocity preservation property of the method. They also demonstrate that the incorporation of the Lagrangian meshing with the MMPDE moving mesh method works well to concentrate mesh points in regions of shocks and material interfaces., 44 pages, 71 figures

The selection of time step plays a crucial role in improving stability and efficiency in the Discontinuous Galerkin (DG) solution of hyperbolic conservation laws on adaptive moving meshes that typically employs explicit stepping. A commonly used selection of time step is a direct extension based on Courant-Friedrichs-Levy (CFL) conditions established for fixed and uniform meshes. In this work, we provide a mathematical justification for those time step selection strategies used in practical adaptive DG computations. A stability analysis is presented for a moving mesh DG method for linear scalar conservation laws. Based on the analysis, a new selection strategy of the time step is proposed, which takes into consideration the coupling of the $\alpha$-function (that is related to the eigenvalues of the Jacobian matrix of the flux and the mesh movement velocity) and the heights of the mesh elements. The analysis also suggests several stable combinations of the choices of the $\alpha$-function in the numerical scheme and in the time step selection. Numerical results obtained with a moving mesh DG method for Burgers' and Euler equations are presented. For comparison purpose, numerical results obtained with an error-based time step-size selection strategy are also given., Comment: 30 pages

A high-order, well-balanced, positivity-preserving quasi-Lagrange moving mesh DG method is presented for the shallow water equations with non-flat bottom topography. The well-balance property is crucial to the ability of a scheme to simulate perturbation waves over the lake-at-rest steady state such as waves on a lake or tsunami waves in the deep ocean. The method combines a quasi-Lagrange moving mesh DG method, a hydrostatic reconstruction technique, and a change of unknown variables. The strategies in the use of slope limiting, positivity-preservation limiting, and change of variables to ensure the well-balance and positivity-preserving properties are discussed. Compared to rezoning-type methods, the current method treats mesh movement continuously in time and has the advantages that it does not need to interpolate flow variables from the old mesh to the new one and places no constraint for the choice of an update scheme for the bottom topography on the new mesh. A selection of one- and two-dimensional examples are presented to demonstrate the well-balance property, positivity preservation, and high-order accuracy of the method and its ability to adapt the mesh according to features in the flow and bottom topography., 42 pages. arXiv admin note: substantial text overlap with arXiv:2006.15187

A fast method is presented for adaptive moving mesh generation in multi-dimensions using a domain decomposition parabolic Monge-Amp\`ere approach. The domain decomposition procedure employed here is non-iterative and involves splitting the computational domain into overlapping subdomains. An adaptive mesh on each subdomain is then computed as the image of the solution of the $L^2$ optimal mass transfer problem using a parabolic Monge-Amp\`ere method. The domain decomposition approach allows straightforward implementation for the parallel computation of adaptive meshes which helps to reduce computational time significantly. Results are presented to show the numerical convergence of the domain decomposition solution to the single domain solution. Several numerical experiments are given to demonstrate the performance and efficiency of the proposed method. The numerical results indicate that the domain decomposition parabolic Monge-Amp\`ere method is more efficient than the standard implementation of the parabolic Monge-Amp\`ere method on the whole domain, in particular when computing adaptive meshes in three spatial dimensions.

Ionic flow carries electrical signals for cells to communicate with each other. The permanent charge of an ion channel is a crucial protein structure for flow properties while boundary conditions play a role of the driving force. Their effects on flow properties have been analyzed via a quasi-one-dimensional Poisson-Nernst-Planck model for small and relatively large permanent charges. The analytical studies have led to the introduction of flux ratios that reflect permanent charge effects and have a universal property. The studies also show that the flux ratios have different behaviors for small and large permanent charges. However, the existing analytical techniques can reveal neither behaviors of flux ratios nor transitions between small and large permanent charges. In this work we present a numerical investigation on flux ratios to bridge between small and large permanent charges. Numerical results verify the analytical predictions for the two extremal regions. More significantly, emergence of non-trivial behaviors is detected as the permanent charge varies from small to large. In particular, saddle-node bifurcations of flux ratios are revealed, showing rich phenomena of permanent charge effects by the power of combining analytical and numerical techniques. An adaptive moving mesh finite element method is used in the numerical studies., 31 pages, 15 figures

There are numerous large-scale applications requiring mesh adaptivity, e.g., computational fluid dynamics and weather prediction. Parallel processing is needed for simulations involving large-scale adaptive meshes. In this paper, we propose a parallel variational mesh quality improvement algorithm for use with distributed memory machines. Our method parallelizes the serial variational mesh quality improvement method by Huang and Kamenski. Their approach is based on the use of the Moving Mesh PDE method to adapt the mesh based on the minimization of an energy functional for mesh equidistribution and alignment. This leads to a system of ordinary differential equations (ODEs) to be solved which determine where to move the interior mesh nodes. An efficient solution is obtained by solving the ODEs on subregions of the mesh with overlapped communication and computation. Strong and weak scaling experiments on up to 128 cores for meshes with up to 160M elements demonstrate excellent results.

A fast method is presented for adaptive moving mesh generation in multi-dimensions using a domain decomposition parabolic Monge-Amp��re approach. The domain decomposition procedure employed here is non-iterative and involves splitting the computational domain into overlapping subdomains. An adaptive mesh on each subdomain is then computed as the image of the solution of the $L^2$ optimal mass transfer problem using a parabolic Monge-Amp��re method. The domain decomposition approach allows straightforward implementation for the parallel computation of adaptive meshes which helps to reduce computational time significantly. Results are presented to show the numerical convergence of the domain decomposition solution to the single domain solution. Several numerical experiments are given to demonstrate the performance and efficiency of the proposed method. The numerical results indicate that the domain decomposition parabolic Monge-Amp��re method is more efficient than the standard implementation of the parabolic Monge-Amp��re method on the whole domain, in particular when computing adaptive meshes in three spatial dimensions.

This article presents an introduction to MMPDElab, a package written in MATLAB for adaptive mesh movement and adaptive moving mesh P1 finite element solution of second-order partial different equations having continuous solutions in one, two, and three spatial dimensions. MMPDElab uses simplicial meshes., 15 pages

The radiative transfer equation models the interaction of radiation with scattering and absorbing media and has important applications in various fields in science and engineering. It is an integro-differential equation involving time, space and angular variables and contains an integral term in angular directions while being hyperbolic in space. The challenges for its numerical solution include the needs to handle with its high dimensionality, the presence of the integral term, and the development of discontinuities and sharp layers in its solution along spatial directions. Its numerical solution is studied in this paper using an adaptive moving mesh discontinuous Galerkin method for spatial discretization together with the discrete ordinate method for angular discretization. The former employs a dynamic mesh adaptation strategy based on moving mesh partial differential equations to improve computational accuracy and efficiency. Its mesh adaptation ability, accuracy, and efficiency are demonstrated in a selection of one- and two-dimensional numerical examples., 32 pages

In the phase-field modeling of brittle fracture, anisotropic constitutive assumptions for the degradation of stored elastic energy due to fracture are crucial to preventing cracking in compression and obtaining physically sound numerical solutions. Three energy decomposition models, the spectral decomposition, the volumetric-deviatoric split, and an improved volumetric-deviatoric split, and their effects on the performance of the phase-field modeling are studied. Meanwhile, anisotropic degradation of stiffness may lead to a small energy remaining on crack surfaces, which violates crack boundary conditions and can cause unphysical crack openings and propagation. A simple yet effective treatment for this is proposed: define a critically damaged zone with a threshold parameter and then degrade both the active and passive energies in the zone. A dynamic mesh adaptation finite element method is employed for the numerical solution of the corresponding elasticity system. Four examples, including two benchmark ones, one with complex crack systems, and one based on an experimental setting, are considered. Numerical results show that the spectral decomposition and improved volumetric-deviatoric split models, together with the improvement treatment of crack boundary conditions, can lead to crack propagation results that are comparable with the existing computational and experimental results. It is also shown that the numerical results are not very sensitive to the parameter defining the critically damaged zone., Comment: 38 pages

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 reference

We study a mesh smoothing algorithm based on the moving mesh PDE (MMPDE) method. For the MMPDE itself, we employ a simple and efficient direct geometric discretization of the underlying meshing functional on simplicial meshes. The nodal mesh velocities can be expressed in a simple, analytical matrix form, which makes the implementation of the method relatively easy and simple. Numerical examples are provided.

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., Improved introduction, added relevant literature references

Preservation of the maximum principle is studied for the combination of the linear finite element method in space and the $\theta$-method in time for solving time dependent anisotropic diffusion problems. It is shown that the numerical solution satisfies a discrete maximum principle when all element angles of the mesh measured in the metric specified by the inverse of the diffusion matrix are non-obtuse and the time step size is bounded below and above by bounds proportional essentially to the square of the maximal element diameter. The lower bound requirement can be removed when a lumped mass matrix is used. In two dimensions, the mesh and time step conditions can be replaced by weaker Delaunay-type conditions. Numerical results are presented to verify the theoretical findings., Comment: 25 pages, 7 figures, 4 tables

The paper presents a numerical study for the finite element method with anisotropic meshes. We compare the accuracy of the numerical solutions on quasi-uniform, isotropic, and anisotropic meshes for a test problem which combines several difficulties of a corner singularity, a peak, a boundary layer, and a wavefront. Numerical experiment clearly shows the advantage of anisotropic mesh adaptation. The conditioning of the resulting linear equation system is addressed as well. In particular, it is shown that the conditioning with adaptive anisotropic meshes is not as bad as generally assumed., 10 pages, 4 figures, revised introduction

Anisotropic mesh adaptation has been successfully applied to the numerical solution of partial differential equations but little considered for variational problems. In this paper, we investigate the use of a global hierarchical basis error estimator for the development of an anisotropic metric tensor needed for the adaptive finite element solution of variational problems. The new metric tensor is completely a~posteriori and based on residual, edge jumps and the hierarchical basis error estimator. Numerical results show that it performs comparable with existing metric tensors based on Hessian recovery. A few sweeps of the symmetric Gau{\ss}-Seidel iteration for solving the global error problem prove sufficient to provide directional information necessary for successful mesh adaptation. ., Comment: 18 pages, 7 figures

The stability of three moving-mesh finite-difference schemes is studied in the L∞ norm for one-dimensional linear convection–diffusion equations. These schemes use central finite differences for spatial discretization and the θ method for temporal discretization, and they are based on conservative and non-conservative forms of transformed partial differential equations. The stability conditions obtained consist of the CFL condition and the mesh speed related conditions. The CFL condition is independent of the mesh speed and has the same form as that for fixed meshes. The mesh speed related conditions restrict how fast the mesh can move. The conditions of this type obtained in this paper are weaker than those in the existing literature and can be satisfied when the mesh is sufficiently fine. Illustrative numerical results are presented.