In this talk, we present two frameworks for constructing finite elements: lattice decomposition and t–n decomposition.
The first part focuses on the construction of finite elements via lattice decomposition. The simplicial lattice is the set of multi-indices with a fixed sum. Existing scalar finite elements can be systematically constructed through different lattice decompositions. In particular, we provide an explicit and computable framework for the smooth finite elements constructed by Hu, Lin, and Wu.
The second part introduces a t–n decomposition for vector and tensor finite elements. At each subsimplex, the space is decomposed into tangential and normal components. Using the t–n frame, we construct decompositions for tensors and differential forms, leading to a new framework for building vector and tensor finite elements.
Combined with lattice decomposition, we obtain a geometric decomposition of finite element spaces and, based on these decompositions, construct finite element complexes. This approach also allows us to use standard Lagrange elements for practical implementation.
This is joint work with Xuehai Huang (Shanghai University of Finance and Economics).