Mathematical Models and Methods in Applied Sciences

We develop higher order multipoint flux mixed finite element (MFMFE) methods for solving elliptic problems on quadrilateral and hexahedral grids that reduce to cell-based pressure systems. The methods are based on a new family of mixed finite elements, which are enhanced Raviart–Thomas spaces with bubbles that are curls of specially chosen polynomials. The velocity degrees of freedom of the new spaces can be associated with the points of tensor-product Gauss–Lobatto quadrature rules, which allows for local velocity elimination and leads to a symmetric and positive definite cell-based system for the pressures. We prove optimal $k$th order convergence for the velocity and pressure in their natural norms, as well as $(k+1)$st order superconvergence for the pressure at the Gauss points. Moreover, local postprocessing gives a pressure that is superconvergent of order $(k+1)$ in the full $L^2$-norm. Numerical results illustrating the validity of our theoretical results are included.

This paper revolves around a newly introduced weak solvability concept for rate-independent systems, alternative to the notions of Energetic ($\mathrm{\text{E}}$) and Balanced Viscosity ($\mathrm{\text{BV}}$) solutions. Visco-Energetic ($\mathrm{\text{VE}}$) solutions have been recently obtained by passing to the time-continuous limit in a time-incremental scheme, akin to that for $\mathrm{\text{E}}$ solutions, but perturbed by a “viscous” correction term, as in the case of $\mathrm{\text{BV}}$ solutions. However, for VE solutions this viscous correction is tuned by a fixed parameter. The resulting solution notion turns out to describe a kind of evolution in between Energetic and BV evolution. In this paper we aim to investigate the application of $\mathrm{\text{VE}}$ solutions to nonsmooth rate-independent processes in solid mechanics such as damage and plasticity at finite strains. We also address the limit passage, in the $\mathrm{\text{VE}}$ formulation, from an adhesive contact to a brittle delamination system. The analysis of these applications reveals the wide applicability of this solution concept, in particular to processes for which $\mathrm{\text{BV}}$ solutions are not available, and confirms its intermediate character between the $\mathrm{\text{E}}$ and $\mathrm{\text{BV}}$ notions.

We consider an initial boundary problem for the planar MHD system under the general condition on the heat conductivity coefficient that depends on both the temperature and the density. Firstly, the global existence of strong solution for large initial data is obtained, and then the limit of the vanishing shear viscosity is justified. In addition, the $L^2$ convergence rate is obtained together with the estimation on the thickness of the boundary layer.

In this paper, we provide a priori error estimates in standard Sobolev (semi-)norms for approximation in spline spaces of maximal smoothness on arbitrary grids. The error estimates are expressed in terms of a power of the maximal grid spacing, an appropriate derivative of the function to be approximated, and an explicit constant which is, in many cases, sharp. Some of these error estimates also hold in proper spline subspaces, which additionally enjoy inverse inequalities. Furthermore, we address spline approximation of eigenfunctions of a large class of differential operators, with a particular focus on the special case of periodic splines. The results of this paper can be used to theoretically explain the benefits of spline approximation under $k$-refinement by isogeometric discretization methods. They also form a theoretical foundation for the outperformance of smooth spline discretizations of eigenvalue problems that has been numerically observed in the literature, and for optimality of geometric multigrid solvers in the isogeometric analysis context.

We deal with flows of non-Newtonian fluids in three-dimensional setting subjected to the homogeneous Dirichlet boundary condition. Under the natural monotonicity, coercivity and growth condition on the Cauchy stress tensor expressed by a power index $p\ge 11/5$, we establish regularity properties of a solution with respect to time variable. Consequently, we can use this better information for showing the uniqueness of the solution provided that the initial data are good enough for all power–law indices $p\ge 11/5$. Such a result was available for $p\ge 12/5$ and therefore the paper fills the gap and extends the uniqueness result to the whole range of $p$’s for which the energy equality holds.