The Mortar Method in the Wavelet Context

Abstract

Abstract. 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 for non conforming domain decomposition, and point out some basic assumptions under which stability and convergence of such method can be proven. We study the application of the mortar method in the biorthogonal wavelet framework. In particular we define suitable multiplier spaces for imposing weak continuity. Unlike in the classical mortar method, such multiplier spaces are not a subset of the space of traces of interior functions, but rather of their duals.For the resulting method, we provide with an error estimate, which is optimal in the geometrically conforming case.Mathematics Subject Classification. 65N55, 42C40, 65N30, 65N15.

Mortar spectral element discretization of the Laplace and Darcy equations with discontinuous coefficients

“…The space L 2 (Ω) 2 is now equipped with the norm 12) and the space H 1 (Ω) with the norm defined in (2.7). First, we have the ellipticity property…”

“…• the following matching condition holds for all m, 12) where the mortar function ϕ(q) associated with q is defined on each γ − m , 1 ≤ m ≤ M − , as the trace of q |Ω − m . The space X is not a discrete space, and its main advantage is that it contains both H 1 (Ω) and all spaces X δ .…”

Basics and some applications of the mortar element method

“…In the case where u is a vector field (e.g., a vector potential), the transmission conditions (43) are written for the either tangential or normal component of u that is physically continuous across S (see [16,51] for more details). To apply a finite element or spectral discretization, we rewrite the eddy current problem in variational form.…”

“…We illustrate this by applications to the finite element discretization, the spectral element discretization and the coupling of both methods. We refer to [42] and [43] for a very interesting work concerning the mortar element method for wavelets, also to [22] and [29] for the use of the mortar element method in the framework of the hp version of the finite element method that we do not consider here and to [3] for mortar finite volumes. Section 3 is devoted to the derivation of a priori error estimates.…”

Coupling Darcy and Stokes equations for porous media with cracks

“…The dependency of α(ε) with respect to ε is only linked to the norm of the extension by zero from H 1 2 −ε (ê) into H 1 2 −ε (∂K) and it is checked in ( [7], Rem. 2.10), that this constant is smaller than c ε −1 .…”