Mathematical Models and Methods in Applied Sciences

The accuracy of the diamond scheme is experimentally investigated for anisotropic diffusion problems in two space dimensions. This finite volume formulation is cell-centered on unstructured triangulations and the numerical method approximates the cell averages of the solution by a suitable discretization of the flux balance at cell boundaries. The key ingredient that allows the method to achieve second-order accuracy is the reconstruction of vertex values from cell averages. For this purpose, we review several techniques from the literature and propose a new variant of the reconstruction algorithm that is based on linear Least Squares. Our formulation unifies the treatment of internal and boundary vertices and includes information from boundaries as linear constraints of the Least Squares minimization process. It turns out that this formulation is well-posed on those unstructured triangulations that satisfy a general regularity condition. The performance of the finite volume method with different algorithms for vertex reconstructions is examined on three benchmark problems having full Dirichlet, Dirichlet-Robin and Dirichlet–Neumann boundary conditions. Comparison of experimental results shows that an important improvement of the accuracy of the numerical solution is attained by using our Least Squares-based formulation. In particular, in the case of Dirichlet–Neumann boundary conditions and strongly anisotropic diffusions the good behavior of the method relies on the absence of locking phenomena that appear when other reconstruction techniques are used.

In this paper, we develop the a posteriori error estimation of hp-version interior penalty discontinuous Galerkin discretizations of elliptic boundary-value problems. Computable upper and lower bounds on the error measured in terms of a natural (mesh-dependent) energy norm are derived. The bounds are explicit in the local mesh sizes and approximation orders. A series of numerical experiments illustrate the performance of the proposed estimators within an automatic hp-adaptive refinement procedure.

We consider a procedure for cancer therapy which consists of injecting replication-competent viruses into the tumor. The viruses infect tumor cells, replicate inside them, and eventually cause their death. As infected cells die, the viruses inside them are released and then proceed to infect adjacent tumor cells. However, a major factor influencing the efficacy of virus agents is the immune response that may limit the replication and spread of the replication-competent virus. The competition between tumor cells, a replication-competent virus and an immune response is modelled as a free boundary problem for a nonlinear system of partial differential equations, where the free boundary is the surface of the tumor. In this model, the immune response equation is a semilinear parabolic equation, including a chemotaxis term which is used to describe the movement of the immune response induced by gradients of the infected cell density. Under the assumption that the chemotactic sensitivity coefficient is small compared with the diffusion coefficient of the immune response, we prove the global existence and uniqueness of the solution of this free boundary problem. For large chemotactic coefficient, the global existence is still open.

In this paper, we address the problem of existence, approximation, and uniqueness of solutions to an abstract doubly nonlinear equation, modeling a rate-independent process with hysteretic behavior. Models of this kind arise in, e.g., plasticity, solid phase transformations, and several other problems in non smooth mechanics. Existence of solutions is proved via passage to the limit in a time-discretization scheme, whereas uniqueness results are obtained by means of convex analysis techniques.

This paper is concerned with the asymptotic behavior of global solutions to a parabolic–hyperbolic coupled system which describes the evolution of the relative temperature θ and the order parameter χ in a material subject to phase transitions. For the system with homogeneous Neumann boundary conditions for both ¸ and χ, under the assumption that the nonlinearities λ and ϕ are real analytic functions, we prove the convergence of a global solution to an equilibrium as time goes to infinity by means of a suitable Łojasiewicz–Simon type inequality.

This paper investigates front propagation in random media for a free boundary problem arising in combustion theory. We show the existence of asymptotic traveling waves solutions with effective speed depending only on the essential infimum of the combustion rate. This result generalizes a previous result of the same authors in the periodic case.