Conforming Finite Element DIVDIV Complexes and the Application for the Linearized Einstein--Bianchi System
Jun Hu, Yizhou Liang, Rui Ma
Abstract
This paper presents the first family of conforming finite element $\ddiv\ddiv$ complexes on tetrahedral grids in three dimensions. In these complexes, finite element spaces of $H(\ddiv\ddiv,\Omega;\Sbb)$ are from a recent article [L. Chen and X. Huang, Math. Comp., 91 (2022), pp. 1107--1142] while finite element spaces of both $H(\sym\ccurl,\Omega;\Tbb)$ and $H^1(\Omega;\R^3)$ are newly constructed here. It is proved that these finite element complexes are exact. As a result, they can be used to discretize the linearized Einstein--Bianchi system within the dual formulation.
Topics & Concepts
Finite element methodOmegaMathematicsDiscretizationExtended finite element methodElement (criminal law)TetrahedronEinsteinMathematical analysisMathematical physicsMixed finite element methodGeometryPure mathematicsPhysicsQuantum mechanicsPolitical scienceLawThermodynamicsAdvanced Numerical Methods in Computational MathematicsNumerical methods for differential equationsElasticity and Material Modeling