Design and Implementation of the Gram-Schmidt Orthogonalization Process in the Sobolev Space H^1 (Ω)

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.