10 research outputs found
A New Triangular Spectral Element Method I: Implementation and Analysis on a Triangle
This paper serves as our first effort to develop a new triangular spectral
element method (TSEM) on unstructured meshes, using the rectangle-triangle
mapping proposed in the conference note [21]. Here, we provide some new
insights into the originality and distinctive features of the mapping, and show
that this transform only induces a logarithmic singularity, which allows us to
devise a fast, stable and accurate numerical algorithm for its removal.
Consequently, any triangular element can be treated as efficiently as a
quadrilateral element, which affords a great flexibility in handling complex
computational domains. Benefited from the fact that the image of the mapping
includes the polynomial space as a subset, we are able to obtain optimal -
and -estimates of approximation by the proposed basis functions on
triangle. The implementation details and some numerical examples are provided
to validate the efficiency and accuracy of the proposed method. All these will
pave the way for developing an unstructured TSEM based on, e.g., the
hybridizable discontinuous Galerkin formulation
On the optimal rates of convergence of Gegenbauer projections
In this paper we present a comprehensive convergence rate analysis of
Gegenbauer projections. We show that, for analytic functions, the convergence
rate of the Gegenbauer projection of degree is the same as that of the best
approximation of the same degree when and the former is slower
than the latter by a factor of when , where
is the parameter in Gegenbauer polynomials. For piecewise analytic functions,
we demonstrate that the convergence rate of the Gegenbauer projection of degree
is the same as that of the best approximation of the same degree when
and the former is slower than the latter by a factor of
when . The extension to functions of fractional
smoothness is also discussed. Our theoretical findings are illustrated by
numerical experiments.Comment: 30 pages; 8 figure
Regularity theory and high order numerical methods for the (1D)-fractional Laplacian
This paper presents regularity results and associated high order numerical methods for one-dimensional fractional-Laplacian boundary-value problems. On the basis of a factorization of solutions as a product of a certain edge-singular weight ω times a "regular" unknown, a characterization of the regularity of solutions is obtained in terms of the smoothness of the corresponding right-hand sides. In particular, for right-hand sides which are analytic in a Bernstein ellipse, analyticity in the same Bernstein ellipse is obtained for the ``regular'' unknown. Moreover, a sharp Sobolev regularity result is presented which completely characterizes the co-domain of the fractional-Laplacian operator in terms of certain weighted Sobolev spaces introduced in (Babuška and Guo, SIAM J. Numer. Anal. 2002). The present theoretical treatment relies on a full eigendecomposition for a certain weighted integral operator in terms of the Gegenbauer polynomial basis. The proposed Gegenbauer-based Nyström numerical method for the fractional-Laplacian Dirichlet problem, further, is significantly more accurate and efficient than other algorithms considered previously. The sharp error estimates presented in this paper indicate that the proposed algorithm is spectrally accurate, with convergence rates that only depend on the smoothness of the right-hand side. In particular, convergence is exponentially fast (resp. faster than any power of the mesh-size) for analytic (resp. infinitely smooth) right-hand sides. The properties of the algorithm are illustrated with a variety of numerical results
Construction of Modern Robust Nodal Discontinuous Galerkin Spectral Element Methods for the Compressible Navier-Stokes Equations
Discontinuous Galerkin (DG) methods have a long history in computational
physics and engineering to approximate solutions of partial differential
equations due to their high-order accuracy and geometric flexibility. However,
DG is not perfect and there remain some issues. Concerning robustness, DG has
undergone an extensive transformation over the past seven years into its modern
form that provides statements on solution boundedness for linear and nonlinear
problems.
This chapter takes a constructive approach to introduce a modern incarnation
of the DG spectral element method for the compressible Navier-Stokes equations
in a three-dimensional curvilinear context. The groundwork of the numerical
scheme comes from classic principles of spectral methods including polynomial
approximations and Gauss-type quadratures. We identify aliasing as one
underlying cause of the robustness issues for classical DG spectral methods.
Removing said aliasing errors requires a particular differentiation matrix and
careful discretization of the advective flux terms in the governing equations.Comment: 85 pages, 2 figures, book chapte