A triangular spectral element method; applications to the incompressible Navier-Stokes equations

Abstract

Encouraged by the success of spectral elements methods in computational fluid dynamics and p-type finite element methods in structural mechanics we wish to extend these ideas to solving high order polynomial approximations on triangular domains as the next generation of spectral element solvers. We introduce here a complete formulation using a modal basis which has been implemented in a new codeNεκTαr. The new basis has the following properties: Jacobi polynomials of mixed weights; semi-orthogonality; hierarchical structure; generalized tensor (warped) product; variable order; and a new apex co-ordinate system allowing automated integration with Gaussian quadrature. We have discussed the formulation using a matrix notation which allows for an easy interpretation of the forward and backward transformations. We use this notation to formulate the linear advection and Helmholtz equations in an efficient manner and show that we recover the following properties: well conditioned matrices; asymptotic operation count of O(N3); scaling O(N2 of the spectral radius of the weak convective operator; and exponential convergence using polynomials up to ∼ 40. Having constructed and numerically analysed these equations we are then able to solve the incompressible Navier-Stokes equations using a high-order splitting scheme. We demonstrate a variety of results showing exponential convergence using both deformed and straight triangular subdomains for both the Stokes and Navier-Stokes problems.