A High-Order Discretization Scheme for Surface Integral Equations for Analyzing the Electroencephalography Forward Problem

TL;DR

This paper introduces a high-order discretization scheme for surface integral equations in EEG forward problem analysis, offering improved flexibility and accuracy over existing methods through novel interpolation point selection and extension to multiple formulations.

Contribution

It presents a novel high-order discretization scheme with flexible basis function order manipulation and extends it to various formulations for enhanced EEG forward problem analysis.

Findings

Demonstrates high accuracy of the scheme in numerical experiments.

Shows increased flexibility and efficiency compared to existing methods.

Validates the approach for multiple surface integral equation formulations.

Abstract

A Nystrom-based high-order (HO) discretization scheme for surface integral equations (SIEs) for analyzing the electroencephalography (EEG) forward problem is proposed in this work. We use HO surface elements and interpolation functions for the discretization of the interfaces of the head volume and the unknowns on the elements, respectively. The advantage of this work over existing isoparametric HO discretization schemes resides in the fact that the interpolation points are different from the mesh nodes, allowing for the flexible manipulation of the order of the basis functions without regenerating the mesh of the interfaces. Moreover, the interpolation points are chosen from the quadrature rules with the same number of points on the elements simplifying the numerical computation of the surface integrals for the far-interaction case. In this contribution, we extend the implementation of the HO discretization scheme to the double-layer and the adjoint double-layer formulations, as well as to the isolated-skull-approach for the double-layer formulation and to the indirect adjoint double-layer formulation, employed to improve the solution accuracy in case of high conductivity contrast models, which requires the development of different techniques for the singularity treatment. Numerical experiments are presented to demonstrate the accuracy, flexibility, and efficiency of the proposed scheme for the four SIEs for analyzing the EEG forward problem.