A multiscale Galerkin method for second-order boundary value problems of Fredholm integro-differential equation II: Efficient algorithm for the discrete linear system

• Published in: Journal of visual communication and image representation 2019-01, Vol.58, p.112-118
• Main Authors: Chen, Jian, He, Minfan, Zeng, Taishan
• Format: Article
• Language: English
• Subjects: Boundary value problems; Fredholm integro-differential equation; Multilevel augmentation method (MAM); Multiscale Galerkin method (MGM)
• ISSN: 1047-3203; 1095-9076
• DOI: 10.1016/j.jvcir.2018.11.027

A multiscale Galerkin method (MGM) was proposed recently by the same authors in order to solve second-order boundary value problems of Fredholm integro-differential equation. Although, the numerical solution of MGM is always stable because of the multiscale bases properties, obligatory of considerable computational cost and huge memory for achieving great approximation accuracy, are the main draw backs. To overcome MGM problems, in this paper, a new multilevel augmentation method (MAM) in order to solve discrete linear system is proposed. Applying the special matrix splitting techniques, approximate solution is obtained by (1) solving a linear system only at an initial lower level; (2) compensating the error by directly computing the product of matrices and vectors at the higher level without any iterations. Theoretical and experimental results approve that MAM and MGM have similar and optimum convergence orders, though MAM is more efficient than MGM.