Analysis and computation of a least-squares method for consistent mesh tying

Abstract

In the finite element method, a standard approach to mesh tying is to apply Lagrange multipliers. If the interface is curved, 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 [T.A. Laursen, M.W. Heinstein, Consistent mesh-tying methods for topologically distinct discretized surfaces in non-linear solid mechanics, Internat. J. Numer. Methods Eng. 57 (2003) 1197-1242].This paper presents a theoretical and computational study of a least-squares method for mesh tying [P. Bochev, D.M. Day, A leastsquares method for consistent mesh tying, Internat. J. Numer. Anal. Modeling 4 (2007) 342-352], applied to the partial differential equation −∇ 2 + = f . We prove optimal convergence rates for domains represented as overlapping subdomains and show 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 we use interface perturbations to eliminate the gaps. Theoretical error estimates are illustrated by numerical experiments.

A stabilized Nitsche overlapping mesh method for the Stokes problem

“…Also, Becker et al [5] presented a Nitsche extended finite element method for incompressible elasticity based on low-order ([P c 1 ] d × P 0 ) elements. Moreover, the least-squares penalty of the velocity differences is related to the mesh tying approach proposed by Day and Bochev [7], who formulate a least-squares problem for a system consisting of the partial differential equation together with the interface conditions. Note that in our method, the interface conditions are enforced using Nitsche's method, while the least-squares terms on the overlap are only included to prove the stability of the method and to control the condition number.…”

A Coupling Approach for Linear Elasticity Problems with Spatially Noncoincident Interfaces

“…The method proposed in [9] removes this complexity by perturbing the meshes at the discrete interfaces so that the area of the gaps and overlaps in the subdomain meshes are equal. [3] removes this perturbation requirement by requiring that the subdomain meshes are only overlapping. Because these methods are based on minimizing a variational principle over the entire problem domain, their applicability is limited to cases where the material constant on both sides of the domain are the same.…”

A Nitsche-based cut finite element method for a fluid-structure interaction problem

“…To achieve a physically more consistent coupling between the solution parts presented on different domains, Schwarz-type domain iteration schemes using Dirichlet/Neumann and Robin coupling on overlapping domains have been proposed for the Navier-Stokes equations in [31]. A completely different route was taken by Day and Bochev [18] who reformulated elliptic interface problems as suitable first-order systems augmented with least-square stabilizations to enforce the interface conditions between the mesh domains to be tied together.…”