Approximation of skewed interfaces with tensor-based model reduction procedures: application to the reduced basis hierarchical model reduction approach

TL;DR

This paper presents a tensor-based model reduction method that effectively approximates solutions with skewed interfaces or strong gradients, improving convergence in PDE solutions.

Contribution

It introduces a novel procedure for locating and removing skewed interfaces, enhancing tensor-based model reduction accuracy for PDEs with complex solution features.

Findings

Accurately locates skewed interfaces using PDE solutions or data functions.

Significantly improves convergence behavior in reduced basis hierarchical model reduction.

Effective even with approximate interface location.

Abstract

In this article we introduce a procedure, which allows to recover the potentially very good approximation properties of tensor-based model reduction procedures for the solution of partial differential equations in the presence of interfaces or strong gradients in the solution which are skewed with respect to the coordinate axes. The two key ideas are the location of the interface either by solving a lower-dimensional partial differential equation or by using data functions and the subsequent removal of the interface of the solution by choosing the determined interface as the lifting function of the Dirichlet boundary conditions. We demonstrate in numerical experiments for linear elliptic equations and the reduced basis-hierarchical model reduction approach that the proposed procedure locates the interface well and yields a significantly improved convergence behavior even in the case when we only consider an approximation of the interface.