Analysis and computation of a least-squares method for consistent mesh tying
In the finite element method, a standard approach to mesh tying is to apply Lagrange multipliers. For curved interfaces, however, discretization generally leads to adjoining surfaces that do not coincide spatially. Straightforward Lagrange multiplier methods lead to discrete formulations failing a first-order patch test as reported by \textit{T. A. Laursen} and \textit{M. W. Heinstein} [Int. J. Numer. Methods Eng. 57, No. 9, 1197--1242 (2003; Zbl 1062.74618)]. This paper presents a theoretical and computational study of a least-squares method for mesh tying, which was formerly proposed by the present authors [Int. J. Numer. Anal. Model. 4, No. 3--4, 342--352 (2007; Zbl 1132.65104)], and applied to the partial differential equation \(-\nabla^2\varphi+ \alpha\varphi= f\), which is transformed into a system of first-order partial differential equations. Optimal convergence rates for domains represented as overlapping subdomains are proven and it is shown that the least-squares method passes a patch test of the order of the finite element space by construction. To apply the method to subdomain configurations with gaps and overlaps, interface perturbations to eliminate the gaps are used. Theoretical error estimates are confirmed by numerical experiments for a two-dimensional sample problem over the unit square domain. Comparison with some other reference numerical results is also made. It appears that the next subjects are to establish the universality of the method and present efficient computational algorithms.
scientific article; zbMATH DE number 5295226
scientific article