Design and Implementation of the Gram-Schmidt Orthogonalization Process in the Sobolev Space H^1 (Ω)
Emilien LORANU LONDJIRINGA *
Exact Sciences Section, Department of Mathematics and Physics, ISP BUNIA, Ituri, DRC.
Camile LIKOTELO BINENE
Department of Mathematics, Faculty of Science and Technology, Statistics and Computer Science, National Pedagogical University, Kinshasa, DRC.
Grace NKWESE MAZONI
Department of Mathematics, Faculty of Science and Technology, Statistics and Computer Science, National Pedagogical University, Kinshasa, DRC.
Jonathan OPFOINTSHI ENGOMBANGI
Higher Institute of Medical Techniques of Bandundu (ISTM BDD), DRC.
Anyta MUKAWA LUKENZU
Department of Mathematics, Faculty of Science and Technology, Statistics and Computer Science, National Pedagogical University, Kinshasa, DRC.
Clara PALUKU KASOKI
Faculty of Science and Technology, Department of Mathematics, Statistics and Computer Science, Université Pédagogique Nationale, Kinshasa, DRC.
*Author to whom correspondence should be addressed.
Abstract
This article presents the design and implementation of the Gram-Schmidt orthogonalization method in the Sobolev space \(H^1\)(\(\Omega\)). However, through computational processes, it becomes clear that the Gram-Schmidt orthogonalization algorithm in \(H^1\)(\(\Omega\)) aims to transform a set of functions into an orthogonal set. By considering an arbitrary basis of a subspace of functions in the \(H^1\)(\(\Omega\)) space, we can construct a new orthogonal basis. However, this method presents certain complexities and Streamlined to improve flow due to the tedious calculation of inner products and norms of the functions. This complexity can lead to an accumulation of errors during the orthogonalization process, thereby compromising the accuracy of the results obtained. The motivation behind the development of the new implementation method is based on the need to reduce the maximum computation time and optimize precision while minimizing the risk of errors during critical steps. Consequently, the improved new approach can not only facilitate the use of this method but also ensure reliable and efficient results in practical applications.
Keywords: Sobolev space H^1 (Ω), dot product in H^1 (Ω), norms in H^1 (Ω), partial derivatives in the sense of distributions, implementation, Algorithm, Lebesgue spaces〖 L〗^2 (Ω), distribution spaces D'(Ω)