Nodal high-order methods on unstructured grids
Journal of Computational Physics
Journal of Computational Physics
Spectral Element Methods on Unstructured Meshes: Comparisons and Recent Advances
Journal of Scientific Computing
A high-order discontinuous Galerkin method for the unsteady incompressible Navier-Stokes equations
Journal of Computational Physics
Journal of Scientific Computing
ELLAM for resolving the kinematics of two-dimensional resistive magnetohydrodynamic flows
Journal of Computational Physics
Journal of Computational Physics
Improved Performance for Nodal Spectral Element Operators
International Journal of High Performance Computing Applications
Polymorphic nodal elements and their application in discontinuous Galerkin methods
Journal of Computational Physics
Higher-order Finite Elements for Hybrid Meshes Using New Nodal Pyramidal Elements
Journal of Scientific Computing
Revisiting and Extending Interface Penalties for Multi-domain Summation-by-Parts Operators
Journal of Scientific Computing
Simulation of multistage excavation based on a 3D spectral-element method
Computers and Structures
High-order optimal edge elements for pyramids, prisms and hexahedra
Journal of Computational Physics
Interior penalty discontinuous Galerkin method on very general polygonal and polyhedral meshes
Journal of Computational and Applied Mathematics
Hi-index | 0.04 |
A framework for the construction of stable spectral methods on arbitrary domains with unstructured grids is presented. Although most of the developments are of a general nature, an emphasis is placed on schemes for the solution of partial differential equations defined on the tetrahedron.In the first part the question of well-behaved multivariate polynomial interpolation on the tetrahedron is addressed, and it is shown how to extend the electrostatic analogy of the Jacobi polynomials to problems beyond the line. This allows for the identification of nodal sets suitable for polynomial interpolation within the tetrahedron and, subsequently, for the formulation of stable spectral schemes on such unstructured nodal sets. The second part of this work is devoted to a discussion of weakly imposed boundary conditions, and energy-stable schemes are formulated for a wide class of problems, exemplified by advection problems, advection-diffusion problems, and linear symmetric hyperbolic systems.Finally, in the third part, issues related to computational efficiency and implementation of the schemes are discussed. The spectral accuracy of the approximation is confirmed through an example, and factorization methods for the efficient computation of derivatives on the general nodal sets within the d-simplex are developed, ensuring that the proposed schemes are competitive with tensor-product-based methods. In this last part we also show that the advective operator results in an ${\cal O}(n^{-2})$ restriction on the time-step, similar to that of spectral collocation methods employing a tensor-product-based approximation. The performance of the proposed scheme is illustrated by solving a wave problem on a triangulated domain, confirming the expected accuracy and stability.