Mortar Methods with Optimized Transmission Conditions for Advection-Diffusion Problems

Abstract

In many practical applications in fluid dynamics, a very large range of scales spanning many orders of magnitude are simultaneously present; one possibility to perform an economical and accurate approximation of the solution is to use different discretizations in different regions of the computational domain to match with the physical scales. The mortar element method introduced in [3] allows such a use of different discretizations in an optimal way in the sense that the error is bounded by the sum of the subregion-by-subregion approximation errors without constraint on the choice of the different discretizations. An extension to fluids is given in [1]. An alternative and simpler method, the New Interface Cement Equilibrated Mortar (NICEM) method proposed in [6] and analyzed in [8] for an elliptic problem, allows to optimally match Robin conditions on non-conforming grids. An extension to Ventcel conditions is given in [9]. The main feature of this approach is that, on each side of the interface, the jump of the Robin or Ventcel condition should be L ²-orthogonal to a well chosen finite element space on the interface (in that case there is no master and slave sides, which makes the method simpler). Thus, it allows to combine different approximations in different subdomains in the framework of optimized Schwarz algorithms which are based on optimized Robin or Ventcel transmission conditions and lead to robust and fast algorithms (see [5, 7]).