Isogeometric solution of Helmholtz equation with Dirichlet boundary condition: numerical experiences

TL;DR

This paper demonstrates the effectiveness of the isogeometric method using B-spline functions for solving the Helmholtz equation with complex boundary conditions, achieving accurate solutions with low computational cost.

Contribution

It introduces a detailed isogeometric approach for Helmholtz problems with nonhomogeneous Dirichlet conditions, including knot selection and refinement strategies for challenging solutions.

Findings

Accurate solutions for Helmholtz problems with oscillatory and discontinuous gradients.

Efficient implementation with low relative error and computational cost.

Effective knot insertion techniques for irregular boundary regions.

Abstract

In this paper we use the Isogeometric method to solve the Helmholtz equation with nonhomogeneous Dirichlet boundary condition over a bounded physical domain. Starting from the variational formulation of the problem, we show with details how to apply the isogeometric approach to obtain an approximation of the solution using biquadratic B-spline functions. To illustrate the power of the method we solve several difficult problems, which are particular cases of the Helmholtz equation, where the solution has discontinuous gradient in some points, or it is highly oscillatory. For these problems we explain how to select the knots of B-spline quadratic functions and how to insert knew knots in order to obtain good approximations of the exact solution on regions with irregular boundary. The results, obtained with our Julia implementation of the method, prove that isogeometric approach produces approximations with a relative small error and computational cost.