Non Uniform Rational B-Splines and Lagrange approximations for time-harmonic acoustic scattering: accuracy and absorbing boundary conditions

TLDR: In this article, the performance of the finite element method based on Lagrange basis functions and the Non Uniform Rational B-Splines (NURBS) based Iso-Geometric Analysis (IGA) are systematically studied for solving time-harmonic acoustic scattering problems.

Abstract: In this paper, the performance of the finite element method based on Lagrange basis functions and the Non Uniform Rational B-Splines (NURBS) based Iso-Geometric Analysis (IGA) are systematically studied for solving time-harmonic acoustic scattering problems. To assess their performance, the numerical examples are presented with truncated absorbing boundary conditions. In the first two examples , we eliminate the domain truncation error by applying second-order Bayliss-Gunzburger-Turkel (BGT-2) Absorbing Boundary Condition (ABC) and modifying the exact solution. Hence, the calculated error is an indicator of the numerical accuracy in the bounded computational domain with no artificial domain truncation error. Next, we apply a higher order local ABC based on the Karp's and Wilcox's far-field expansions for 2D and 3D problems, respectively. The performance of both methods in solving exterior problems is compared. The introduced auxiliary surface functions are also estimated using the corresponding basis functions. The influence of various parameters, viz., order of the approximating polynomial, number of degrees of freedom, wave number and the boundary conditions (BGT-2 and number of terms in the far-field expansions) on the accuracy and convergence rate is systematically studied. It is inferred that, irrespective of the order of the polynomial, IGA yields higher accuracy per degree of freedom when compared to the conventional finite element method with Lagrange basis.