Numerical Analysis

Fluid-structure interaction The nonlinear coupling of the equations governing fluid-structure interaction systems requires appropriate numerical approaches in order to deal with the motion of the domains occupied by solid and fluid. One of the main issues is the construction of stable numerical schemes. Two different approaches have been considered: Immersed boundary method Arbitrary Lagrangian-Eulerian formulation Approximation of PDEs by finite element methods The finite element method is one of the most popular methods available for the numerical resolution of PDEs of different types. In view of practical applications, the finite elements methods need to be robust, efficient, and accurate. In the case of finite element for problems in mixed form, this requires that some compatibility conditions are satisfied. Finite element methods for the approximation of eigenproblem in mixed form Finite element approximation of evolution problem in mixed form Edge finite elements for Maxwell and photonic crystal equations Finite elements for the Stokes problem Domain Decomposition Methods (DDM) for Heterogeneous Problems Subdomain splitting is an interesting path towards multiphysics (or heterogeneous problems) in which different kinds of PDE (modeling different physical phenomena) are set up in different subdomains. Examples are the coupling of Stokes equations with the Darcy equations to simulate the filtration of fluids in porous media; the coupling between advection-diffusion with dominated advection and pure advection phenomena; the coupling between the Navier-Stokes equation and the system of linear or nonlinear elasticity for fluid-structure interactions. Interface Control Domain Decomposition (ICDD) methods are overlapping DDM that are well suited to face heterogeneous problems. INTERNODES: a general-purpose method to deal with non-conforming discretizations of partial differential equations on regions partitioned into two or several disjoint subdomains. It exploits two intergrid interpolation operators, one for transferring the Dirichlet trace across the interfaces, the others for the Neumann trace. High-order methods for the approximation of PDE's Spectral Methods are high order methods for solving PDE's which offer the best performance (in terms of computational efficiency and in handling complex geometries) when they are coupled with either low-order methods (such as finite elements) inside the preconditioning step, and domain decomposition techniques. Algebraic Fractional-Step Schemes are very efficient and accurate techniques to approximate time-dependent PDE's as, e.g., the incompressible Navier-Stokes equations. Finite-element preconditioning of spectral methods Algebraic fractional step schemes for the incompressible Navier-Stokes equations