Analysis of the Shifted Boundary Method for the Poisson Problem in General Domains

TL;DR

This paper provides a comprehensive analysis of the shifted boundary method (SBM) for the Poisson problem, demonstrating optimal convergence rates and stability in both smooth and non-smooth domains, with insights into geometric influences.

Contribution

It offers a rigorous analysis of SBM's convergence, stability, and error behavior in complex geometries, including procedures for boundary shifting and conditions for well-posedness.

Findings

Optimal convergence in energy norm.

Enhanced convergence in L2 norm.

Influence of geometry on accuracy and stability.

Abstract

The shifted boundary method (SBM) is an approximate domain method for boundary value problems, in the broader class of unfitted/embedded/immersed methods. It has proven to be quite efficient in handling problems with complex geometries, ranging from Poisson to Darcy, from Navier-Stokes to elasticity and beyond. The key feature of the SBM is a {\it shift} in the location where Dirichlet boundary conditions are applied - from the true to a surrogate boundary - and an appropriate modification (again, a {\it shift}) of the value of the boundary conditions, in order to reduce the consistency error. In this paper we provide a sound analysis of the method in smooth and non-smooth domains, highlighting the influence of geometry and distance between exact and surrogate boundaries upon the convergence rate. Without loss of generality, we consider the Poisson problem with Dirichlet boundary conditions as a model and we first detail a procedure to obtain the crucial shifting between the surrogate and the true boundaries. Next, we give a sufficient condition for the well-posedness and stability of the discrete problem. The behavior of the consistency error arising from shifting the boundary conditions is thoroughly analyzed, for smooth boundaries and for boundaries with corners and edges. The convergence rate is proven to be optimal in the energy norm, and is further enhanced in the $L^2$-norm.