2019/10/29 by Jeonghun J. Lee, Lee, Jeonghun J. · 1 citation
Engineering · Mathematics · #65N30 #Advanced Numerical Methods in Computational Mathematics #Differential Equations and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods for differential equations
paper · pdf · doi:10.48550/arxiv.1910.13059
openalex publication_date 2019/10/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In mixed finite element approximations of Hodge Laplace problems associated with the de Rham complex, the exterior derivative operators are computed exactly, so the spatial locality is preserved. However, the numerical approximations of the associated coderivatives are nonlocal and it can be regarded as an undesired effect of standard mixed methods. For numerical methods with local coderivatives a perturbation of low order mixed methods in the sense of variational crimes has been developed for simplicial and cubical meshes. In this paper we extend the low order method to all high orders on cubical meshes using a new family of finite element differential forms on cubical meshes. The key theoretical contribution is a generalization of the linear degree, in the construction of the serendipity family of differential forms, and the generalization is essential in the unisolvency proof of the new family of finite element differential forms.