Orthogonalization in high-order finite element method

Abstract

The finite element method employs local basis functions to build the approximation field within each finite element. These functions, known as shape functions, must be tailored to the shapes of the elements while maintaining the global conformity of the approximation. Shape functions in finite elements are categorized into two types: bubble functions and edge functions. Bubble functions are zero on the element’s boundary, while edge functions are the remaining ones. The degrees of freedom (dofs) across the entire mesh consist of inner dofs within each element, skeleton dofs at the element edges inside the domain, and boundary dofs at the domain’s outer boundary. When the bilinear form of the considered problem is Hermitian and coercive in $H_0^1$, it can be interpreted as the inner product. This paper utilizes such an inner product to construct orthogonal bubble functions and edge functions orthogonal to the bubble functions. Consequently, the problem matrix in each element is partially diagonal, allowing the inner degrees of freedom to be determined within each element before the assembly process, and the global problem only involves the skeleton and boundary dofs. In the subsequent step, the skeleton basis functions are orthogonalized, reducing the global problem to only the boundary degrees of freedom. This method can be effectively applied to analyze a collection of problems with varying boundary conditions. The orthogonalization process combines the generalized eigenvalue problem and Gaussian elimination, stored in a square matrix, to switch between standard and orthogonalized shape functions or degrees of freedom. A substantial amount of calculations occurs within the finite elements, making them inherently suitable for parallel processing. This technique, referred to as the orthogonalized FEM (OFEM), is suitable for high-order finite elements to reduce memory usage, simplify the assembly procedure, and can greatly decrease computation time when executed in parallel. Several 2D examples, including the Poisson problem, stationary and non-stationary heat flow, and linear elasticity, demonstrate the proposed approach’s effectiveness.