Gap-SBM: A New Conceptualization of the Shifted Boundary Method with Optimal Convergence for the Neumann and Dirichlet Problems

TL;DR

This paper introduces a new Shifted Boundary Method that achieves optimal accuracy for boundary value problems by accurately modeling the boundary gap and extending solutions, validated through extensive 2D tests.

Contribution

The paper develops a novel Shifted Boundary Method with a three-stage process for improved boundary condition treatment and provides rigorous mathematical analysis of its optimal convergence.

Findings

Achieves optimal $L^2$- and $H^1$-norm error convergence

Effectively models boundary gaps with a new approximation technique

Validated through extensive 2D numerical experiments

Abstract

We propose and mathematically analyze a new Shifted Boundary Method for the treatment of Dirichlet and Neumann boundary conditions, with provable optimal accuracy in the $L^2$- and $H^1$-norms of the error. The proposed method is built on three stages. First, the distance map between the SBM surrogate boundary and the true boundary is used to construct an approximation to the geometry of the gap between the two. Then, the representations of the numerical solution and test functions are extended from the surrogate domain to such gap. Finally, approximate quadrature formulas and specific shift operators are applied to integrate a variational formulation that also involves the fields extended in the gap. An extensive set of two-dimensional tests demonstrates the theoretical findings and the overall optimal performance of the proposed method.