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 moving mesh partial differential equation approach and employs its newly developed implementation. The implementation has several improvements over the traditional one, including its explicit, compact form of the mesh velocities, ease to program, and less likelihood of producing singular meshes. Three types of metric tensor that correspond to uniform and arclength-based and Hessian-based adaptive meshes are considered. The method shows first-order convergence for uniform and arclength-based adaptive meshes, and 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.

An alternating direction approximate Newton (ADAN) method is developed for solving inverse problems of the form min{φ(Bu) + (1/2) Au − f 2 2 }, where φ is convex and possibly nonsmooth, and A and B are matrices. Problems of this form arise in image reconstruction where A is the matrix describing the imaging device, f is the measured data, φ is a regularization term, and B is a derivative operator. The proposed algorithm is designed to handle applications where A is a large dense, ill-conditioned matrix. The algorithm is based on the alternating direction method of multipliers (ADMM) and an approximation to Newton's method in which a term in Newton's Hessian is replaced by a Barzilai-Borwein (BB) approximation. It is shown that ADAN converges to a solution of the inverse problem. Numerical results are provided using test problems from parallel magnetic resonance imaging. ADAN was faster than a proximal ADMM scheme that does not employ a BB Hessian approximation, while it was more stable and much simpler than the related Bregman operator splitting algorithm with variable stepsize algorithm which also employs a BB-based Hessian approximation.

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.

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.