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Finite element discretization of Stokes problems can result in singular, inconsistent saddle point linear algebraic systems. This inconsistency can cause many iterative methods to fail to converge. In this work, we consider the lowest-order weak Galerkin finite element method to discretize Stokes flow problems and study a consistency enforcement by modifying the right-hand side of the resulting linear system. It is shown that the modification of the scheme does not affect the optimal-order convergence of the numerical solution. Moreover, inexact block diagonal and triangular Schur complement preconditioners and the minimal residual method (MINRES) and the generalized minimal residual method (GMRES) are studied for the iterative solution of the modified scheme. Bounds for the eigenvalues and the residual of MINRES/GMRES are established. Those bounds show that the convergence of MINRES and GMRES is independent of the viscosity parameter and mesh size. The convergence of the modified scheme and effectiveness of the preconditioners are verified using numerical examples in two and three dimensions., Comment: arXiv admin note: text overlap with arXiv:2409.16477

This work is concerned with the convergence of the iterative solution for the Stokes flow, discretized with the weak Galerkin finite element method and preconditioned using inexact block Schur complement preconditioning. The resulting saddle point linear system is singular and the pressure solution is not unique. The system is regularized with a commonly used strategy by specifying the pressure value at a specific location. It is analytically shown that the regularized system is nonsingular but has an eigenvalue approaching zero as the fluid kinematic viscosity tends to zero. Inexact block diagonal and triangular Schur complement preconditioners are considered with the minimal residual method (MINRES) and the generalized minimal residual method (GMRES), respectively. For both cases, the bounds are obtained for the eigenvalues of the preconditioned systems and for the residual of MINRES/GMRES. These bounds show that the convergence factor of MINRES/GMRES is almost independent of the viscosity parameter and mesh size while the number of MINRES/GMRES iterations required to reach convergence depends on the parameters only logarithmically. The theoretical findings and effectiveness of the preconditioners are verified with two- and three-dimensional numerical examples., Comment: 22 pages

In the finite difference approximation of the fractional Laplacian the stiffness matrix is typically dense and needs to be approximated numerically. The effect of the accuracy in approximating the stiffness matrix on the accuracy in the whole computation is analyzed and shown to be significant. Four such approximations are discussed. While they are shown to work well with the recently developed grid-over finite difference method (GoFD) for the numerical solution of boundary value problems of the fractional Laplacian, they differ in accuracy, economics to compute, performance of preconditioning, and asymptotic decay away from the diagonal line. In addition, two preconditioners based on sparse and circulant matrices are discussed for the iterative solution of linear systems associated with the stiffness matrix. Numerical results in two and three dimensions are presented., Comment: 28 pages

A moving mesh finite element method is studied for the numerical solution of Bernoulli free boundary problems. The method is based on the pseudo-transient continuation with which a moving boundary problem is constructed and its steady-state solution is taken as the solution of the underlying Bernoulli free boundary problem. The moving boundary problem is solved in a split manner at each time step: the moving boundary is updated with the Euler scheme, the interior mesh points are moved using a moving mesh method, and the corresponding initial-boundary value problem is solved using the linear finite element method. The method can take full advantages of both the pseudo-transient continuation and the moving mesh method. Particularly, it is able to move the mesh, free of tangling, to fit the varying domain for a variety of geometries no matter if they are convex or concave. Moreover, it is convergent towards steady state for a broad class of free boundary problems and initial guesses of the free boundary. Numerical examples for Bernoulli free boundary problems with constant and non-constant Bernoulli conditions and for nonlinear free boundary problems are presented to demonstrate the accuracy and robustness of the method and its ability to deal with various geometries and nonlinearities., Comment: 26 pages

A so-called grid-overlay finite difference method (GoFD) was proposed recently for the numerical solution of homogeneous Dirichlet boundary value problems of the fractional Laplacian on arbitrary bounded domains. It was shown to have advantages of both finite difference and finite element methods, including its efficient implementation through the fast Fourier transform and ability to work for complex domains and with mesh adaptation. The purpose of this work is to study GoFD in a meshfree setting, a key to which is to construct the data transfer matrix from a given point cloud to a uniform grid. Two approaches are proposed, one based on the moving least squares fitting and the other based on the Delaunay triangulation and piecewise linear interpolation. Numerical results obtained for examples with convex and concave domains and various types of point clouds are presented. They show that both approaches lead to comparable results. Moreover, the resulting meshfree GoFD converges at a similar order as GoFD with unstructured meshes and finite element approximation as the number of points in the cloud increases. Furthermore, numerical results show that the method is robust to random perturbations in the location of the points., Comment: 18 pages

A grid-overlay finite difference method is proposed for the numerical approximation of the fractional Laplacian on arbitrary bounded domains. The method uses an unstructured simplicial mesh and an overlay uniform grid for the underlying domain and constructs the approximation based on a uniform-grid finite difference approximation and a data transfer from the unstructured mesh to the uniform grid. The method takes full advantages of both uniform-grid finite difference approximation in efficient matrix-vector multiplication via the fast Fourier transform and unstructured meshes for complex geometries and mesh adaptation. It is shown that its stiffness matrix is similar to a symmetric and positive definite matrix and thus invertible if the data transfer has full column rank and positive column sums. Piecewise linear interpolation is studied as a special example for the data transfer. It is proved that the full column rank and positive column sums of linear interpolation is guaranteed if the spacing of the uniform grid is smaller than or equal to a positive bound proportional to the minimum element height of the unstructured mesh. Moreover, a sparse preconditioner is proposed for the iterative solution of the resulting linear system for the homogeneous Dirichlet problem of the fractional Laplacian. Numerical examples demonstrate that the new method has similar convergence behavior as existing finite difference and finite element methods and that the sparse preconditioning is effective. Furthermore, the new method can readily be incorporated with existing mesh adaptation strategies. Numerical results obtained by combining with the so-called MMPDE moving mesh method are also presented., Comment: 27 pages

A well-balanced moving mesh discontinuous Galerkin (DG) method is proposed for the numerical solution of the Ripa model -- a generalization of the shallow water equations that accounts for effects of water temperature variations. Thermodynamic processes are important particularly in the upper layers of the ocean where the variations of sea surface temperature play a fundamental role in climate change. The well-balance property which requires numerical schemes to preserve the lake-at-rest steady state is crucial to the simulation of perturbation waves over that steady state such as waves on a lake or tsunami waves in the deep ocean. To ensure the well-balance, positivity-preserving, and high-order properties, a DG-interpolation scheme (with or without scaling positivity-preserving limiter) and special treatments pertaining to the Ripa model are employed in the transfer of both the flow variables and bottom topography from the old mesh to the new one and in the TVB limiting process. Mesh adaptivity is realized using an MMPDE moving mesh approach and a metric tensor based on an equilibrium variable and water depth. A motivation is to adapt the mesh according to both the perturbations of the lake-at-rest steady state and the water depth distribution (bottom structure). Numerical examples in one and two dimensions are presented to demonstrate the well-balance, high-order accuracy, and positivity-preserving properties of the method and its ability to capture small perturbations of the lake-at-rest steady state., Comment: 37 pages, 24 figures. arXiv admin note: substantial text overlap with arXiv:2006.15187

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., Comment: 13 pages

Adaptive moving spatial meshes are useful for solving physical models given by time-dependent partial differentialequations. However, special consideration must be given when combining adaptive meshing procedures with ensemble-based data assimilation (DA) techniques. In particular, we focus on the case where each ensemble member evolvesindependently upon its own mesh and is interpolated to a common mesh for the DA update. This paper outlines aframework to develop time-dependent reference meshes using locations of observations and the metric tensors (MTs)or monitor functions that define the spatial meshes of the ensemble members. We develop a time-dependent spatiallocalization scheme based on the metric tensor (MT localization). We also explore how adaptive moving mesh tech-niques can control and inform the placement of mesh points to concentrate near the location of observations, reducingthe error of observation interpolation. This is especially beneficial when we have observations in locations that wouldotherwise have a sparse spatial discretization. We illustrate the utility of our results using discontinuous Galerkin(DG) approximations of 1D and 2D inviscid Burgers equations. The numerical results show that the MT localizationscheme compares favorably with standard Gaspari-Cohn localization techniques. In problems where the observationsare sparse, the choice of common mesh has a direct impact on DA performance. The numerical results also demonstratethe advantage of DG-based interpolation over linear interpolation for the 2D inviscid Burgers equation.

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 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., Comment: 44 pages, 71 figures

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., Comment: 42 pages. arXiv admin note: substantial text overlap with arXiv:2006.15187

A rezoning-type adaptive moving mesh discontinuous Galerkin method is proposed for the numerical solution of the shallow water equations with non-flat bottom topography. The well-balance property is crucial to the simulation of perturbation waves over the lake-at-rest steady state such as waves on a lake or tsunami waves in the deep ocean. To ensure the well-balance and positivity-preserving properties, strategies are discussed in the use of slope limiting, positivity-preservation limiting, and data transferring between meshes. Particularly, it is suggested that a DG-interpolation scheme be used for the interpolation of both the flow variables and bottom topography from the old mesh to the new one and after each application of the positivity-preservation limiting on the water depth, a high-order correction be made to the approximation of the bottom topography according to the modifications in the water depth. Moreover, mesh adaptation based on the equilibrium variable and water depth is shown to give more desirable results than that based on the commonly used entropy function. Numerical examples in one and two spatial dimensions are presented to demonstrate the well-balance and positivity-preserving properties of the method and its ability to capture small perturbations of the lake-at-rest steady state., Comment: 49 pages

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., Comment: 31 pages, 15 figures

Solution interpolation between deforming meshes is an important component for several applications in scientific computing, including indirect arbitrary-Lagrangian-Eulerian and rezoning moving mesh methods in numerical solution of partial differential equations. In this paper, a high-order, conservative, and positivity-preserving interpolation scheme is developed based on the discontinuous Galerkin solution of a linear time-dependent equation on deforming meshes. The scheme works for bounded but otherwise arbitrary mesh deformation from the old mesh to the new one. The cost and positivity preservation (with a linear scaling limiter) of the DG-interpolation are investigated. Numerical examples are presented to demonstrate the properties of the interpolation scheme. The DG-interpolation is applied to the rezoning moving mesh DG solution of the radiative transfer equation, an integro-differential equation modeling the conservation of photons and involving time, space, and angular variables. Numerical results obtained for examples in one and two spatial dimensions with various settings show that the resulting rezoning moving mesh DG method maintains the same convergence order as the standard DG method, is more efficient than the method with a fixed uniform mesh, and is able to preserve the positivity of the radiative intensity., Comment: 28 pages

We study the stability of explicit Runge-Kutta methods for high order Lagrangian finite element approximation of linear parabolic equations and establish bounds on the largest eigenvalue of the system matrix which determines the largest permissible time step. A bound expressed in terms of the ratio of the diagonal entries of the stiffness and mass matrices is shown to be tight within a small factor which depends only on the dimension and the choice of the reference element and basis functions but is independent of the mesh or the coefficients of the initial-boundary value problem under consideration. Another bound, which is less tight and expressed in terms of mesh geometry, depends only on the number of mesh elements and the alignment of the mesh with the diffusion matrix. The results provide an insight into how the interplay between the mesh geometry and the diffusion matrix affects the stability of explicit integration schemes when applied to a high order finite element approximation of linear parabolic equations on general nonuniform meshes., Comment: \copyright 2015 Springer International Publishing Switzerland. The final version is available at https://doi.org/10.1007/978-3-319-10705-9_16

A new anisotropic mesh adaptation strategy for finite element solution of elliptic differential equations is presented. It generates anisotropic adaptive meshes as quasi-uniform ones in some metric space, with the metric tensor being computed based on hierarchical a posteriori error estimates. A global hierarchical error estimate is employed in this study to obtain reliable directional information of the solution. Instead of solving the global error problem exactly, which is costly in general, we solve it iteratively using the symmetric Gauss--Seidel (GS) method. Numerical results show that a few GS iterations are sufficient for obtaining a reasonably good approximation to the error for use in anisotropic mesh adaptation. The new method is compared with several strategies using local error estimators or recovered Hessians. Numerical results are presented for a selection of test examples and a mathematical model for heat conduction in a thermal battery with large orthotropic jumps in the material coefficients., Comment: 23 pages, 10 figures, 1 table. \copyright 2009. Licensed under CC-BY-NC-ND 4.0 (https://creativecommons.org/licenses/by-nc-nd/4.0)

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., Comment: 15 pages

A surface moving mesh method is presented for general surfaces with or without explicit parameterization. The method can be viewed as a nontrivial extension of the moving mesh partial differential equation method that has been developed for bulk meshes and demonstrated to work well for various applications. The main challenges in the development of surface mesh movement come from the fact that the Jacobian matrix of the affine mapping between the reference element and any simplicial surface element is not square. The development starts with revealing the relation between the area of a surface element in the Euclidean or Riemannian metric and the Jacobian matrix of the corresponding affine mapping, formulating the equidistribution and alignment conditions for surface meshes, and establishing a meshing energy function based on the conditions. The moving mesh equation is then defined as the gradient system of the energy function, with the nodal mesh velocities being projected onto the underlying surface. The analytical expression for the mesh velocities is obtained in a compact, matrix form, which makes the implementation of the new method on a computer relatively easy and robust. Moreover, it is analytically shown that any mesh trajectory generated by the method remains nonsingular if it is so initially. It is emphasized that the method is developed directly on surface meshes, making no use of any information on surface parameterization. It utilizes surface normal vectors to ensure that the mesh vertices remain on the surface while moving, and also assumes that the initial surface mesh is given. The new method can apply to general surfaces with or without explicit parameterization since the surface normal vectors can be computed even when the surface only has a numerical representation. A selection of two- and three-dimensional examples are presented., Comment: 39 pages, 13 figures

A moving mesh discontinuous Galerkin method is presented for the numerical solution of hyperbolic conservation laws. The method is a combination of the discontinuous Galerkin method and the mesh movement strategy which is based on the moving mesh partial differential equation approach and moves the mesh continuously in time and orderly in space. It discretizes hyperbolic conservation laws on moving meshes in the quasi-Lagrangian fashion with which the mesh movement is treated continuously and no interpolation is needed for physical variables from the old mesh to the new one. Two convection terms are induced by the mesh movement and their discretization is incorporated naturally in the DG formulation. Numerical results for a selection of one- and two-dimensional scalar and system conservation laws are presented. It is shown that the moving mesh DG method achieves the theoretically predicted order of convergence for problems with smooth solutions and is able to capture shocks and concentrate mesh points in non-smooth regions. Its advantage over uniform meshes and its insensitiveness to mesh smoothness are also demonstrated., Comment: 41 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., Comment: 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

Numerical and analytical methods are developed for the investigation of contact sets in electrostatic-elastic deflections modeling micro-electro mechanical systems. The model for the membrane deflection is a fourth-order semi-linear partial differential equation and the contact events occur in this system as finite time singularities. Primary research interest is in the dependence of the contact set on model parameters and the geometry of the domain. An adaptive numerical strategy is developed based on a moving mesh partial differential equation to dynamically relocate a fixed number of mesh points to increase density where the solution has fine scale detail, particularly in the vicinity of forming singularities. To complement this computational tool, a singular perturbation analysis is used to develop a geometric theory for predicting the possible contact sets. The validity of these two approaches are demonstrated with a variety of test cases., Comment: 26 pages

A moving mesh finite difference method based on the moving mesh partial differential equation is proposed for the numerical solution of the 2T model for multi-material, non-equilibrium radiation diffusion equations. The model involves nonlinear diffusion coefficients and its solutions stay positive for all time when they are positive initially. Nonlinear diffusion and preservation of solution positivity pose challenges in the numerical solution of the model. A coefficient-freezing predictor-corrector method is used for nonlinear diffusion while a cutoff strategy with a positive threshold is used to keep the solutions positive. Furthermore, a two-level moving mesh strategy and a sparse matrix solver are used to improve the efficiency of the computation. Numerical results for a selection of examples of multi-material non-equilibrium radiation diffusion show that the method is capable of capturing the profiles and local structures of Marshak waves with adequate mesh concentration. The obtained numerical solutions are in good agreement with those in the existing literature. Comparison studies are also made between uniform and adaptive moving meshes and between one-level and two-level moving meshes., Comment: 29 pages

The conditioning of the linear finite volume element discretization for general diffusion equations is studied on arbitrary simplicial meshes. The condition number is defined as the ratio of the maximal singular value of the stiffness matrix to the minimal eigenvalue of its symmetric part. This definition is motivated by the fact that the convergence rate of the generalized minimal residual method for the corresponding linear systems is determined by the ratio. An upper bound for the ratio is established by developing an upper bound for the maximal singular value and a lower bound for the minimal eigenvalue of the symmetric part. It is shown that the bound depends on three factors, the number of the elements in the mesh, the mesh nonuniformity measured in the Euclidean metric, and the mesh nonuniformity measured in the metric specified by the inverse diffusion matrix. It is also shown that the diagonal scaling can effectively eliminates the effects from the mesh nonuniformity measured in the Euclidean metric. Numerical results for a selection of examples in one, two, and three dimensions are presented.

A lack of regularity in the solution of the porous medium equation poses a serious challenge in its theoretical and numerical studies. A common strategy in theoretical studies is to utilize the pressure formulation of the equation where a new variable called the mathematical pressure is introduced. It is known that the new variable has much better regularity than the original one and Darcy's law for the movement of the free boundary can be expressed naturally in this new variable. The pressure formulation has not been used in numerical studies. The goal of this work is to study its use in the adaptive finite element solution of the porous medium equation. The MMPDE moving mesh strategy is employed for adaptive mesh movement while linear finite elements are used for spatial discretization. The free boundary is traced explicitly by integrating Darcy's law with the Euler scheme. Numerical results are presented for three two-dimensional examples. The method is shown to be second-order in space and first-order in time in the pressure variable. Moreover, the convergence order of the error in the location of the free boundary is almost second-order in the maximum norm. However, numerical results also show that the convergence order of the method in the original variable stays between first-order and second-order in the $L^1$ norm or between 0.5th-order and first-order in the $L^2$ norm. Nevertheless, the current method can offer some advantages over numerical methods based on the original formulation for situations with large exponents or when a more accurate location of the free boundary is desired., Comment: 19 pages

The Ambrosio-Tortorelli functional is a phase-field approximation of the Mumford-Shah functional that has been widely used for image segmentation. The approximation has the advantages of being easy to implement, maintaining the segmentation ability, and $\Gamma$-converging to the Mumford-Shah functional. However, it has been observed in actual computation that the segmentation ability of the Ambrosio-Tortorelli functional varies significantly with different values of the parameter and it even fails to $\Gamma$-converge to the original functional for some cases. In this paper we present an asymptotic analysis on the gradient flow equation of the Ambrosio-Tortorelli functional and show that the functional can have different segmentation behavior for small but finite values of the regularization parameter and eventually loses its segmentation ability as the parameter goes to zero when the input image is treated as a continuous function. This is consistent with the existing observation as well as the numerical examples presented in this work. A selection strategy for the regularization parameter and a scaling procedure for the solution are devised based on the analysis. Numerical results show that they lead to good segmentation of the Ambrosio-Tortorelli functional for real images., Comment: 22 pages

A moving mesh finite element method is studied for the numerical solution of a phase-field model for brittle fracture. The moving mesh partial differential equation approach is employed to dynamically track crack propagation. Meanwhile, the decomposition of the strain tensor into tensile and compressive components is essential for the success of the phase-field modeling of brittle fracture but results in a non-smooth elastic energy and stronger nonlinearity in the governing equation. This makes the governing equation much more difficult to solve and, in particular, Newton's iteration often to fail to converge. Three regularization methods are proposed to smooth out the decomposition of the strain tensor. Numerical examples of fracture propagation under quasi-static load demonstrate that all of the methods can effectively improve the convergence of Newton's iteration for relatively small values of the regularization parameter but without comprising the accuracy of the numerical solution. They also show that the moving mesh finite element method is able to adaptively concentrate the mesh elements around propagating cracks and handle multiple and complex crack systems., Comment: 32 pages

The conditioning of implicit Runge-Kutta (RK) integration for linear finite element approximation of diffusion equations on general anisotropic meshes is investigated. Bounds are established for the condition number of the resulting linear system with and without diagonal preconditioning for the implicit Euler and general implicit RK methods. Two solution strategies are considered for the linear system resulting from general implicit RK integration: the simultaneous solution (the system is solved as a whole) and a successive solution which follows the commonly used implementation of implicit RK methods to first transform the system into smaller systems using the Jordan normal form of the RK matrix and then solve them successively. For the simultaneous solution in case of a positive semidefinite symmetric part of the RK coefficient matrix and for the successive solution it is shown that . If the smallest eigenvalue of the symmetric part of the RK coefficient matrix is negative and the simultaneous solution strategy is used, an upper bound on the time step is given so that the system matrix is positive definite. The obtained bounds for the condition number have explicit geometric interpretations and take the interplay between the diffusion matrix and the mesh geometry into full consideration. They show that there are three mesh-dependent factors that can affect the conditioning: the number of elements, the mesh nonuniformity measured in the Euclidean metric, and the mesh nonuniformity with respect to the inverse of the diffusion matrix. They also reveal that the preconditioning using the diagonal of the system matrix, the mass matrix, or the lumped mass matrix can effectively eliminate the effects of the mesh nonuniformity measured in the Euclidean metric. Numerical examples are given., Comment: Accepted manuscript. \copyright 2019. Licensed under CC-BY-NC-ND 4.0 (https://creativecommons.org/licenses/by-nc-nd/4.0)

A third-order moving mesh cell-centered scheme without the remapping of physical variables is developed for the numerical solution of one-dimensional elastic-plastic flows with the Mie-Gr\"{u}neisen equation of state, the Wilkins constitutive model, and the von Mises yielding criterion. The scheme combines the Lagrangian method with the MMPDE moving mesh method and adaptively moves the mesh to better resolve shock and other types of waves while preventing the mesh from crossing and tangling. It can be viewed as a direct arbitrarily Lagrangian-Eulerian method but can also be degenerated to a purely Lagrangian scheme. It treats the relative velocity of the fluid with respect to the mesh as constant in time between time steps, which allows high-order approximation of free boundaries. A time dependent scaling is used in the monitor function to avoid possible sudden movement of the mesh points due to the creation or diminishing of shock and rarefaction waves or the steepening of those waves. A two-rarefaction Riemann solver with elastic waves is employed to compute the Godunov values of the density, pressure, velocity, and deviatoric stress at cell interfaces. Numerical results are presented for three examples. The third-order convergence of the scheme and its ability to concentrate mesh points around shock and elastic rarefaction waves are demonstrated. The obtained numerical results are in good agreement with those in literature. The new scheme is also shown to be more accurate in resolving shock and rarefaction waves than an existing third-order cell-centered Lagrangian scheme., Comment: 26 pages

A new functional is presented for variational mesh generation and adaptation. It is formulated based on combining the equidistribution and alignment conditions into a single condition with only one dimensionless parameter. The functional is shown to be coercive but not convex. A solution procedure using a discrete moving mesh partial differential equation is employed. It is shown that the element volumes and altitudes of a mesh trajectory of the mesh equation associated with the new functional are bounded away from zero and the mesh trajectory stays nonsingular if it is so initially. Numerical examples demonstrate that the new functional performs comparably as an existing one that is also based on the equidistribution and alignment conditions and known to work well but contains an additional parameter.

Preservation of nonnengativity and boundedness in the finite element solution of Nagumo-type equations with general anisotropic diffusion is studied. Linear finite elements and the backward Euler scheme are used for the spatial and temporal discretization, respectively. An explicit, an implicit, and two hybrid explicit-implicit treatments for the nonlinear reaction term are considered. Conditions for the mesh and the time step size are developed for the numerical solution to preserve nonnegativity and boundedness. The effects of lumping of the mass matrix and the reaction term are also discussed. The analysis shows that the nonlinear reaction term has significant effects on the conditions for both the mesh and the time step size. Numerical examples are given to demonstrate the theoretical findings., Comment: 28 pages, 7 figures, 6 tables

An adaptive moving mesh finite element method is proposed for the numerical solution of the regularized long wave (RLW) equation. A moving mesh strategy based on the so-called moving mesh PDE is used to adaptively move the mesh to improve computational accuracy and efficiency. The RLW equation represents a class of partial differential equations containing spatial-time mixed derivatives. For the numerical solution of those equations, a $C^0$ finite element method cannot apply directly on a moving mesh since the mixed derivatives of the finite element approximation may not be defined. To avoid this difficulty, a new variable is introduced and the RLW equation is rewritten into a system of two coupled equations. The system is then discretized using linear finite elements in space and the fifth-order Radau IIA scheme in time. A range of numerical examples in one and two dimensions, including the RLW equation with one or two solitary waves and special initial conditions that lead to the undular bore and solitary train solutions, are presented. Numerical results demonstrate that the method has a second order convergence and is able to move and adapt the mesh to the evolving features in the solution., Comment: 27 pages, 13 figures

An adaptive moving mesh finite element method is studied for the numerical solution of the porous medium equation with and without variable exponents and absorption. The method is based on the so-called moving mesh partial differential equation approach and employs its newly developed implementation. Three types of mesh are considered, uniform and arclength-based and Hessian-based adaptive meshes. The method shows a first order convergence for uniform and arclength-based adaptive meshes and a second-order convergence for Hessian-based adaptive meshes. It is also shown that the method can be used for situations with complex free boundaries, emerging and splitting of free boundaries, and the porous medium equation with variable exponents and absorption. Two dimensional numerical results are presented.

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 Preprint

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 preprint

A hybrid LDG-HWENO scheme is proposed for the numerical solution of KdV-type partial differential equations. It evolves the cell averages of the physical solution and its moments (a feature of Hermite WENO) while discretizes high order spatial derivatives using the local DG method. The new scheme has the advantages of both LDG and HWENO methods, including the ability to deal with high order spatial derivatives and the use of a small number of global unknown variables. The latter is independent of the order of the scheme and the spatial order of the underlying differential equations. One and two dimensional numerical examples are presented to show that the scheme can attain the same formal high order accuracy as the LDG method.

Anisotropic mesh adaptation is studied for linear finite element solution of 3D anisotropic diffusion problems. The M-uniform mesh approach is used, where an anisotropic adaptive mesh is generated as a uniform one in the metric specified by a tensor. In addition to mesh adaptation, preservation of the maximum principle is also studied. Some new sufficient conditions for maximum principle preservation are developed, and a mesh quality measure is defined to server as a good indicator. Four different metric tensors are investigated: one is the identity matrix, one focuses on minimizing an error bound, another one on preservation of the maximum principle, while the fourth combines both. Numerical examples show that these metric tensors serve their purposes. Particularly, the fourth leads to meshes that improve the satisfaction of the maximum principle by the finite element solution while concentrating elements in regions where the error is large. Application of the anisotropic mesh adaptation to fractured reservoir simulation in petroleum engineering is also investigated, where unphysical solutions can occur and mesh adaptation can help improving the satisfaction of the maximum principle., Comment: 30 pages, 18 figures

Anisotropic mesh quality measures and anisotropic mesh adaptation are studied for polygonal meshes. Three sets of alignment and equidistribution measures are developed, one based on least squares fitting, one based on generalized barycentric mapping, and the other based on singular value decomposition of edge matrices. Numerical tests show that all three sets of mesh quality measures provide good measurements for the quality of polygonal meshes under given metrics. Based on one of these sets of quality measures and using a moving mesh partial differential equation, an anisotropic adaptive polygonal mesh method is constructed for the numerical solution of second order elliptic equations. Numerical examples are presented to demonstrate the effectiveness of the method., Comment: 43 figures

Nonnegative directional splittings of anisotropic diffusion operators in the divergence form are investigated. Conditions are established for nonnegative directional splittings to hold in a neighborhood of an arbitrary interior point. The result is used to construct monotone finite difference schemes for the boundary value problem of anisotropic diffusion operators. It is shown that such a monotone scheme can be constructed if the underlying diffusion matrix is continuous on the closure of the physical domain and symmetric and uniformly positive definite on the domain, the mesh spacing is sufficiently small, and the size of finite difference stencil is sufficiently large. An upper bound for the stencil size is obtained, which is determined completely by the diffusion matrix. Loosely speaking, the more anisotropic the diffusion matrix is, the larger stencil is required. An exception is the situation with a strictly diagonally dominant diffusion matrix where a three-by-three stencil is sufficient for the construction of a monotone finite difference scheme. Numerical examples are presented to illustrate the theoretical findings., Comment: 21 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