Spectral methods on triangles and other domains
Journal of Scientific Computing
The Runge-Kutta discontinuous Galerkin method for conservation laws V multidimensional systems
Journal of Computational Physics
Bounds for eigenvalues and condition numbers in the p-version of the finite element method
Mathematics of Computation
Runge–Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems
Journal of Scientific Computing
Locally divergence-free discontinuous Galerkin methods for the Maxwell equations
Journal of Computational Physics
Implementation of hierarchical bases in FEMLAB for simplicial elements
ACM Transactions on Mathematical Software (TOMS)
Uniformly Convergent Iterative Methods for Discontinuous Galerkin Discretizations
Journal of Scientific Computing
A Posteriori Error Control for a Weakly Over-Penalized Symmetric Interior Penalty Method
Journal of Scientific Computing
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We construct well-conditioned orthonormal hierarchical bases for simplicial $\mathcal{L}_{2}$ finite elements. The construction is made possible via classical orthogonal polynomials of several variables. The basis functions are orthonormal over the reference simplicial elements in two and three dimensions. The mass matrices M are identity while the conditioning of the stiffness matrices S grows as $\mathcal{O}(p^{3})$ with respect to the order p. The diagonally normalized stiffness matrices are well conditioned. The diagonally normalized composite matrices 驴M+S are also well conditioned for a wide range of 驴. For the mass, stiffness and composite matrices, the bases in this study have much better conditioning than existing high-order hierarchical bases.