Anisotropic mesh adaptation is studied for linear finite element solution of 3D anisotropic diffusion problems. The -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.
A mesh condition is developed for linear finite element approximations of anisotropic diffusion–convection–reaction problems to satisfy a discrete maximum principle. Loosely speaking, the condition ...requires that the mesh be simplicial and
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-nonobtuse when the dihedral angles are measured in the metric specified by the inverse of the diffusion matrix, where
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perturbation of a generalized Delaunay condition). These results include many existing mesh conditions as special cases. Numerical results are presented to verify the theoretical findings.
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.
A moving collocation method has been shown to be very effcient for the adaptive solution of second- and fourth-order time-dependent partial differential equations and forms the basis for the two ...robust codes MOVCOL and MOVCOL4. In this paper, the relations between the method and the traditional collocation and finite volume methods are investigated. It is shown that the moving collocation method inherits desirable properties of both methods： the ease of implementation and high-order convergence of the traditional collocation method and the mass conservation of the finite volume method. Convergence of the method in the maximum norm is proven for general linear two-point boundary value problems. Numerical results are given to demonstrate the convergence order of the method.
The pressure formulation of the porous medium equation has been commonly used in theoretical studies due to its much better regularities than the original formulation. The goal here is to study its ...use in the adaptive moving mesh finite element solution. The free boundary is traced explicitly through Darcy's law. The method is shown numerically 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 in the original variable stays between first‐order and second‐order in L1 norm or between 0.5th‐order and first‐order in L2 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.
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 depends on the number of mesh elements and corresponds to the classic bound for the Laplace operator on a uniform mesh. The second 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.
Adaptive moving spatial meshes are useful for solving physical models given by time-dependent partial differential equations. 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 evolves independently upon its own mesh and is interpolated to a common mesh for the DA update. This paper outlines a framework 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 spatial localization scheme based on the metric tensor (MT localization). We also explore how adaptive moving mesh techniques can control and inform the placement of mesh points to concentrate near the location of observations, reducing the error of observation interpolation. This is especially beneficial when we have observations in locations that would otherwise 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 localization scheme compares favorably with standard Gaspari-Cohn localization techniques. In problems where the observations are sparse, the choice of common mesh has a direct impact on DA performance. The numerical results also demonstrate the advantage of DG-based interpolation over linear interpolation for the 2D inviscid Burgers equation.
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.
This paper considers several moving mesh partial differential equations that are related to the equidistribution principle. Several of these are new, and some correspond to discrete moving mesh ...equations that have been used by others. Their stability is analyzed and it is seen that a key term for most of these moving mesh PDEs is a source-like term that measures the level of equidistribution. It is shown that under weak assumptions mesh crossing cannot occur for most of them. Finally, numerical experiments for these various moving mesh PDEs are performed to study their relative properties.
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