In this work, we study numerical methods for a coupled fluid-porous media flow model. The model consists of Stokes equations and Darcy's equations in two neighboring subdomains, coupling together through certain interface conditions. The weak form for the coupled model is of saddle point type. A mortar finite element method is proposed to approximate the weak form of the coupled problem. In our method, nonconforming Crouzeix-Raviart elements are applied in the fluid subdomain and the lowest order Raviart-Thomas elements are applied in the porous media subdomain; Meshes in different subdomains are allowed to be nonmatching on the common interface; Interface conditions are weakly imposed via adding constraint in the definition of the finite element space. The well-posedness of the discrete problem and the optimal error estimate for the proposed method are established. Numerical experiments are also given to confirm the theoretical results.

We consider the Laplace equation posed in a three-dimensional axisymmetric domain. Wereduce the original problem by a Fourier expansion in the angular variable to a countablefamily of two-dimensional problems. We decompose the meridian domain, assumed polygonal,in a finite number of rectangles and we discretize by a spectral method. Then we describethe main features of the mortar method and use the algorithm Strang Fix to improve theaccuracy of our discretization.

A multilevel preconditioner based on the abstract framework of theauxiliary space method, is developed for the mortar method for thenonconforming P 1 finite element or the lowest orderCrouzeix-Raviart finite element on nonmatching grids. It is shownthat the proposed preconditioner is quasi-optimal in the sense thatthe condition number of the preconditioned system is independent ofthe mesh size, and depends only quadratically on the number ofrefinement levels. Some numerical results confirming the theory arealso provided.

As a first draft of a model for a river flowing on a homogeneous porous ground, we consider a system where the Darcy and Stokes equations are coupled via appropriate matching conditions on the interface. We propose a discretization of this problem which combines the mortar method with standard finite elements, in order to handle separately the flow inside and outside the porous medium. We prove a priori and a posteriori error estimates for the resulting discrete problem. Some numerical experiments confirm the interest of the discretization.

This paper deals with the mortar spectral element discretization of two equivalent problems, the Laplace equation and the Darcy system, in a domain which corresponds to a nonhomogeneous anisotropic medium. The numerical analysis of the discretization leads to optimal error estimates and the numerical experiments that we present enable us to verify its efficiency.

The reduced basis element method is a new approach for approximating the solution of problems described by partial differential equations. The method takes its roots in domain decomposition methods and reduced basis discretizations. The basic idea is to first decomposethe computational domain into a series of subdomains that are deformationsof a few reference domains (or generic computational parts). Associated with each reference domain are precomputed solutions corresponding to the same governing partial differential equation,but solved for different choices of deformations of the referencesubdomains and mapped onto the reference shape.The approximation corresponding to a new shape is then taken to be a linear combination of the precomputed solutions, mappedfrom the generic computational part to the actual computational part.We extend earlier work in this direction to solve incompressible fluid flow problems governed by the steady Stokes equations. Particular focus is given to satisfying the inf-sup condition,to a posteriori error estimation, and to “gluing” the local solutions together in the multidomain case.

This paper deals with the use of wavelets in the framework of the Mortar method.We first review in an abstract framework the theory of the mortar method fornon conforming domain decomposition, and point out some basic assumptionsunder which stability and convergence of such method can be proven. We studythe application of the mortar method in the biorthogonal wavelet framework.In particular we define suitable multiplier spaces for imposing weakcontinuity. Unlike in the classical mortar method, such multiplier spaces arenot a subset of the space of traces of interior functions, but rather oftheir duals.For the resulting method, we provide with an error estimate, which is optimal in thegeometrically conforming case.

We present an improved, near-optimal hp error estimate for a non-conforming finite element method, called the mortar method (M0). We also present a new hp mortaring technique, called the mortar method (MP), and derive h, p and hp error estimates for it, in the presence of quasiuniform and non-quasiuniform meshes. Our theoretical results, augmented by the computational evidence we present, show that like (M0), (MP) is also a viable mortaring technique for the hp method.