Arnoldi-based orthonormal and hierarchical divergence-free polynomial basis and its applications

Abstract: This paper presents a methodology to construct a divergence-free polynomial basis of an arbitrary degree in a simplex (triangles in 2D and tetrahedra in 3D) of arbitrary dimension. It allows for fast computation of all numerical solutions from degree zero to a specified degree \textit{k} for certain PDEs. The generated divergence-free basis is orthonormal, hierarchical, and robust in finite-precision arithmetic. At the core is an Arnoldi-based procedure. It constructs an orthonormal and hierarchical basis for multi-dimensional polynomials of degree less than or equal to \textit{k}. The divergence-free basis is generated by combining these polynomial basis functions. An efficient implementation of the hybridized BDM mixed method is developed using these basis functions. Hierarchy allows for incremental construction of the global matrix and the global vector for all degrees (zero to \textit{k}) using the local problem solution computed just for degree \textit{k}. Orthonormality and divergence-free properties simplify the local problem. PDEs considered are Helmholtz, Laplace, and Poisson problems in smooth domains and in a corner domain. These advantages extend to other PDEs such as incompressible Stokes, incompressible Navier-Stokes, and Maxwell equations.