Semi-smooth Newton methods for elliptic equations with gradient constraints are investigated. The one- and multi-dimensional cases are treated separately. Numerical examples illustrate the approach and as well as structural features of the solution.

This study concerns some asymptotic models used to compute the flow outside and inside fractures in a bidimensional porous medium. The flow is governed by the Darcy law both in the fractures and in the porous matrix with large discontinuities in the permeability tensor. These fractures are supposed to have a small thickness with respect to the macroscopic length scale,so that we can asymptotically reduce them to immersed polygonal faultinterfaces and the model finally consists in a coupling between a2D elliptic problem and a 1D equation on the sharp interfaces modelling the fractures.A cell-centered finite volume scheme on general polygonal meshes fitting the interfacesis derived to solve the set of equations with the additional differential transmission conditions linking both pressure and normal velocityjumps through the interfaces.We prove the convergence of the FV scheme for any set of data and parameters of the models and derive existence and uniqueness of the solution to the asymptotic models proposed. The models are then numerically experimented for highly or partially immersed fractures.Some numerical results are reported showing different kinds of flowsin the case of impermeable or partially/highly permeable fractures. The influence of the variation of the aperture of the fractures is also investigated. The numericalsolutions of the asymptotic models are validated by comparing them to the solutions of the global Darcy model or to some analytic solutions.

We developed a mimetic finite difference method for solving elliptic equations with tensor coefficients on polyhedral meshes. The first-order convergence estimates in a mesh-dependent H1 norm are derived.

We are concerned with a finite element approximation for time-harmonic wave propagation governed by the Helmholtz equation. The usually oscillatory behavior of solutions, along with numerical dispersion, render standard finite element methods grossly inefficient already in medium-frequency regimes. As an alternative, methods that incorporate information about the solution in the form of plane waves have been proposed. We focus on a class of Trefftz-type discontinuous Galerkin methods that employs trial and test spaces spanned by local plane waves. In this paper we give a priori convergence estimates for the h-version of these plane wave discontinuous Galerkin methods in two dimensions. To that end, we develop new inverse and approximation estimates for plane waves and use these in the context of duality techniques. Asymptotic optimality of the method in a mesh dependent norm can be established. However, the estimates require a minimal resolution of the mesh beyond what it takes to resolve the wavelength. We give numerical evidence that this requirement cannot be dispensed with. It reflects the presence of numerical dispersion.

We present a new semi-discrete central scheme for one-dimensional shallow water flows along channels with non-uniform rectangular cross sections and bottom topography. The scheme preserves the positivity of the water height, and it is preserves steady-states of rest (i.e., it is well-balanced). Along with a detailed description of the scheme, numerous numerical examples are presented for unsteady and steady flows. Comparison with exact solutions illustrate the accuracy and robustness of the numerical algorithm.

We analyze Euler-Galerkin approximations (conforming finite elements inspace and implicit Euler in time) to coupled PDE systems in which one dependentvariable, say u, is governed by an elliptic equation and the other,say p, by a parabolic-like equation. The underlying application is theporoelasticity system within the quasi-static assumption. Differentpolynomial orders are used for the u- and p-components toobtain optimally convergent a priori bounds for allthe terms in the error energy norm.Then, a residual-type a posteriori error analysis is performed. Upon extending theideas of Verfürth for the heat equation [Calcolo40 (2003)195–212],an optimally convergent bound is derived for the dominant term in theerror energy norm and an equivalence result between residual anderror is proven. The error bound can be classically split intotime error, space error and data oscillation terms. Moreover, upon extending the elliptic reconstruction techniqueintroduced by Makridakis and Nochetto [SIAM J. Numer. Anal.41 (2003) 1585–1594],an optimally convergent bound is derived for the remaining terms in theerror energy norm. Numerical results are presented toillustrate the theoretical analysis.

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.