Mathematical Models and Methods in Applied Sciences

We discuss a first-order Cucker–Smale-type consensus model with attractive and repulsive interactions and present upper and lower bound estimates on the number of asymptotic point-clusters depending on the relative ranges of interactions and coupling strength. When the number of agents approaches infinity, we introduce a scalar conservation law with a non-local flux for a macroscopic description. We show that the corresponding conservation law admits a classical solution for sufficiently smooth initial data, which illustrates the shock avoidance effect due to the non-locality of the interactions. We also study the dynamics of special Dirac-Comb-type solutions consisting of two and three point-clusters.

We propose a relaxation framework for general fluid models which can be understood as a natural extension of the Suliciu approach in the Euler setting. In particular, the relaxation system may be totally degenerate. Several stability properties are proved. The relaxation procedure is shown to be efficient in the numerical approximation of the entropy weak solutions of the original PDEs. The numerical method is particularly simple in the case of a fully degenerate relaxation system for which the solution of the Riemann problem is explicit. Indeed, the Godunov solver for the homogeneous relaxation system results in an HLLC-type solver for the equilibrium model. Discrete entropy inequalities are established under a natural Gibbs principle.

We prove weighted anisotropic analytic estimates for solutions of second-order elliptic boundary value problems in polyhedra. The weighted analytic classes which we use are the same as those introduced by Guo in 1993 in view of establishing exponential convergence for hp finite element methods in polyhedra. We first give a simple proof of the known weighted analytic regularity in a polygon, relying on a new formulation of elliptic a priori estimates in smooth domains with analytic control of derivatives. The technique is based on dyadic partitions near the corners. This technique can successfully be extended to polyhedra, providing isotropic analytic regularity. This is not optimal, because it does not take advantage of the full regularity along the edges. We combine it with a nested open set technique to obtain the desired three-dimensional anisotropic analytic regularity result. Our proofs are global and do not require the analysis of singular functions.

We consider mixed hp-discontinuous Galerkin FEM for linear elasticity in polyhedral domains Ω ⊂ ℝ³. In order to resolve possible corner, edge, and corner–edge singularities, anisotropic axiparallel geometric edge meshes consisting of hexahedral elements are applied. We show inf–sup stability results on both the continuous and the discrete level which are robust with respect to the Poisson ratio as it tends to the incompressible limit of ½. Furthermore, in the subsequent a priori error analysis we derive a quasi-optimality result, including the case of singular solutions. In addition, under certain realistic assumptions (for analytic data) on the regularity of the exact solution, we prove that the proposed DG schemes converge at an exponential rate in terms of the fifth root of the number of degrees of freedom.

We prove that solutions of a mildly regularized Perona–Malik equation converge, in a slow time scale, to solutions of the total variation flow. The convergence result is global-in-time, and holds true in any space dimension. The proof is based on the general principle that "the limit of gradient-flows is the gradient-flow of the limit". To this end, we exploit a general result relating the Gamma-limit of a sequence of functionals to the limit of the corresponding maximal slope curves.