The study of heterogeneous physical systems composed of two or more media separated by selective interfaces is a topic of utmost relevance in biology, materials science, nanoelectronics and geophysics, to name a few. The present talk focuses on a class of mathematical problems directly motivated by the aforementioned applications. More precisely, we consider a stationary advection-diffusion-reaction problem in a three-dimensional volume Ω, whose physical properties may vary in space, thereby leading to an elliptic second-order partial differential equation with variable coefficients. In addition, we account for the presence of a selective internal interface Γ, which is geometrically represented by a two-dimensional manifold in Ω and on which we impose transmission conditions that ensure the balance of flux density across the interface and model segregation phenomena that may occur within the interface itself.
To accurately capture the interface phenomena mentioned above, in this talk we propose a novel numerical approach that combines, for the first time:
1- a Dual Mixed Hybrid (DMH) finite element (FE) method to ensure that (i) the solution (or primal variable) verifies the given partial differential equation within each element; (ii) the normal component of the flux density (or dual variable) associated with the solution is continuous across elements; and (iii) both primal and dual variables satisfy optimal error estimates;
2- a Three-Field formulation, typical of domain decomposition approaches, to account for interfacial discontinuities within the weak formulation of the problem;
3- a Streamline Upwind/Petrov-Galerkin (SUPG) stabilization method to gain the required amount of numerical stability without significantly spoiling the accuracy of the computed solution due to excessive crosswind smearing.
We analyze the proposed stabilized DMH FE scheme at both the infinite and finite dimensional levels using the abstract theory of saddle-point problems and we prove its well-posedness and optimal error estimates under suitable assumptions on the data. In addition, we introduce a static condensation procedure to eliminate variables defined in the interior of each element in favor of the sole hybrid variable, thereby obtaining a final algebraic system of much reduced size structurally analogous to that of a standard primal-based finite element approach. Then, we illustrate a series of simulations to validate the accuracy and robustness of the novel method via comparison between numerical and analytical solutions in three-dimensional test cases. Results show that the proposed stabilized DMH FEM method (i) satisfies the theoretical findings even in the presence of marked interface jump discontinuities in the solution and its associated flux; and (ii) is capable of accurately resolving steep boundary and/or interior layers without introducing spurious unphysical oscillations or excessive smearing of the solution front.
Coffee and Cookies will be served in Room 306 at 3:00pm.