In the paper, we present an efficient two-grid method for the approximation of two-dimensional nonlinear reaction-diffusion equations using a expanded mixed finite-element method. We transfer the nonlinear reaction diffusion equation into first order nonlinear equations. The solution of the nonlinear system on the fine space is reduced to the solutions of two small (one linear and one non-linear) systems on the coarse space and a linear system on the fine space. Moreover, we obtain the error estimation for the two-grid algorithm. It is showed that coarse space can be extremely coarse and achieve asymptotically optimal approximation as long as the mesh sizes satisfy . An numerical example is also given to illustrate the effectiveness of the algorithm.

We introduce and analyze a fully-mixed finite element method for a fluid-solidinteraction problem in 2D. The model consists of an elastic body which is subject to agiven incident wave that travels in the fluid surrounding it. Actually, the fluid issupposed to occupy an annular region, and hence a Robin boundary condition imitating thebehavior of the scattered field at infinity is imposed on its exterior boundary, which islocated far from the obstacle. The media are governed by the elastodynamic and acousticequations in time-harmonic regime, respectively, and the transmission conditions are givenby the equilibrium of forces and the equality of the corresponding normal displacements.We first apply dual-mixed approaches in both domains, and then employ the governingequations to eliminate the displacement u of the solid and the pressure pof the fluid. In addition, since both transmission conditions become essential, they areenforced weakly by means of two suitable Lagrange multipliers. As a consequence, theCauchy stress tensor and the rotation of the solid, together with the gradient ofp and the traces of u and p on the boundary of thefluid, constitute the unknowns of the coupled problem. Next, we show that suitabledecompositions of the spaces to which the stress and the gradient of pbelong, allow the application of the Babuška–Brezzi theory and the Fredholm alternativefor analyzing the solvability of the resulting continuous formulation. The unknowns of thesolid and the fluid are then approximated by a conforming Galerkin scheme defined in termsof PEERS elements in the solid, Raviart–Thomas of lowest order in the fluid, andcontinuous piecewise linear functions on the boundary. Then, the analysis of the discretemethod relies on a stable decomposition of the corresponding finite element spaces andalso on a classical result on projection methods for Fredholm operators of index zero.Finally, some numerical results illustrating the theory are presented.

After setting a mixed formulation for the propagation of linearized water waves problem, we define its spectral element approximation. Then, in order to take into account unbounded domains, we construct absorbing perfectly matched layer for the problem. We approximate these perfectly matched layer by mixed spectral elements and show their stability using the “frozen coefficient” technique. Finally, numerical results will prove the efficiency of the perfectly matched layer compared to classical absorbing boundary conditions.

We investigate unilateral contact problems with cohesive forces, leading tothe constrained minimization of a possibly nonconvex functional. Weanalyze the mathematical structure of the minimization problem. The problem is reformulated in terms of a three-field augmentedLagrangian, and sufficient conditions for the existence of a localsaddle-point are derived. Then, we derive and analyze mixed finiteelement approximations to the stationarity conditions of the three-fieldaugmented Lagrangian. The finite element spaces for the bulk displacement andthe Lagrange multiplier must satisfy a discrete inf-sup condition, whilediscontinuous finite element spaces spanned by nodal basis functions areconsidered for the unilateral contact variable so as to use collocationmethods. Two iterative algorithms are presented and analyzed, namely anUzawa-type method within a decomposition-coordination approach and anonsmooth Newton's method. Finally, numerical results illustrating thetheoretical analysis are presented.

The problem of modeling acoustic waves scattered by an object with Neumann boundary condition is considered. The boundary condition is taken into account by means of the fictitious domain method, yielding a first order in time mixed variational formulation for the problem. The resulting system is discretized with two families of mixed finite elements that are compatible with mass lumping. We present numerical results illustrating that the Neumann boundary condition on the object is not always correctly taken into account when the first family of mixed finite elements is used. We, therefore, introduce the second family of mixed finite elements for which a theoretical convergence analysis is presented and error estimates are obtained. A numerical study of the convergence is also considered for a particular object geometry which shows that our theoretical error estimates are optimal.

In this paper we combine the dual-mixed finite element method with a Dirichlet-to-Neumann mapping(given in terms of a boundary integral operator) to solve linear exterior transmission problems inthe plane. As a model we consider a second order elliptic equation in divergence form coupled withthe Laplace equation in the exterior unbounded region. We show that the resulting mixed variationalformulation and an associated discrete scheme using Raviart-Thomas spaces are well posed, and derivethe usual Cea error estimate and the corresponding rate of convergence. In addition, we develop twodifferent a-posteriori error analyses yielding explicit residual and implicit Bank-Weiser typereliable estimates, respectively. Several numerical results illustrate the suitability of theseestimators for the adaptive computation of the discrete solutions.