Convergence of finite element right-hand-side computation from finite difference data

TL;DR

This paper introduces two advanced integration methods for transferring fields from finite difference to finite element frameworks in aeroacoustics, analyzing their accuracy, performance, and convergence properties.

Contribution

It presents novel high-order Gaussian quadrature and supermesh integration techniques that improve accuracy and reduce errors in finite element field transfer from finite difference data.

Findings

High-order quadrature converges with increasing B-Spline order.

Supermesh approach eliminates interpolation errors.

Methods perform well with smooth and oscillating fields.

Abstract

This work presents two integration methods for field transfer in computational aeroacoustics and in coupled field problems, using the finite element method to solve the acoustic field. Firstly, a high-order Gaussian quadrature computes the finite element right-hand side. In contrast, the (flow) field provided by the finite difference mesh is mapped by higher-order B-Splines or a Lagrangian function. Secondly, the cut-cell or supermesh integration with geometric clipping. For each method, the accuracy, performance characteristics, and computational complexity are analyzed. As a reference, the trapezoidal integration rule was computed from the finite difference results. The high-order quadrature converges as the B-Spline interpolation order increases, and the finite difference results and mesh resolutions are consistent. The supermesh approach eliminates interpolation and approximation errors at the grid-to-mesh level and improves accuracy. This behaviour is universal for smooth or strongly oscillating field quantities, which will be shown in a comparative study between the Lighthill-like source term and the source term of the perturbed convective wave equation for subsonic flows.