271 research outputs found
Exploiting Superconvergence Through Smoothness-Increasing Accuracy-Conserving (SIAC) Filtering
There has been much work in the area of superconvergent error analysis for finite element and discontinuous Galerkin (DG) methods. The property of superconvergence leads to the question of how to exploit this information in a useful manner, mainly through superconvergence extraction. There are many methods used for superconvergence extraction such as projection, interpolation, patch recovery and B-spline convolution filters. This last method falls under the class of Smoothness-Increasing Accuracy-Conserving (SIAC) filters. It has the advantage of improving both smoothness and accuracy of the approximation. Specifically, for linear hyperbolic equations it can improve the order of accuracy of a DG approximation from k + 1 to 2k + 1, where k is the highest degree polynomial used in the approximation, and can increase the smoothness to k − 1. In this article, we discuss the importance of overcoming the mathematical barriers in making superconvergence extraction techniques useful for applications, specifically focusing on SIAC filtering
Local discontinuous Galerkin methods for fractional ordinary differential equations
This paper discusses the upwinded local discontinuous Galerkin methods for
the one-term/multi-term fractional ordinary differential equations (FODEs). The
natural upwind choice of the numerical fluxes for the initial value problem for
FODEs ensures stability of the methods. The solution can be computed element by
element with optimal order of convergence in the norm and
superconvergence of order at the downwind point of each
element. Here is the degree of the approximation polynomial used in an
element and () represents the order of the one-term
FODEs. A generalization of this includes problems with classic 'th-term
FODEs, yielding superconvergence order at downwind point as
. The underlying mechanism of the
superconvergence is discussed and the analysis confirmed through examples,
including a discussion of how to use the scheme as an efficient way to evaluate
the generalized Mittag-Leffler function and solutions to more generalized
FODE's.Comment: 17 pages, 7 figure
Superconvergence and \u3ci\u3ea posteriori\u3c/i\u3e error estimates of a local discontinuous Galerkin method for the fourth-order initial-boundary value problems arising in beam theory
In this paper, we investigate the superconvergence properties and a posteriori error estimates of a local discontinuous Galerkin (LDG) method for solving the one-dimensional linear fourth-order initial-boundary value problems arising in study of transverse vibrations of beams. We present a local error analysis to show that the leading terms of the local spatial discretization errors for the k-degree LDG solution and its spatial derivatives are proportional to (k + 1)-degree Radau polynomials. Thus, the k-degree LDG solution and its derivatives are O(hk+2) superconvergent at the roots of (k + 1)-degree Radau polynomials. Computational results indicate that global superconvergence holds for LDG solutions. We discuss how to apply our superconvergence results to construct efficient and asymptotically exact a posteriori error estimates in regions where solutions are smooth. Finally, we present several numerical examples to validate the superconvergence results and the asymptotic exactness of our a posteriori error estimates under mesh refinement. Our results are valid for arbitrary regular meshes and for Pk polynomials with k ≥ 1, and for various types of boundary conditions
Two types of spectral volume methods for 1-D linear hyperbolic equations with degenerate variable coefficients
In this paper, we analyze two classes of spectral volume (SV) methods for
one-dimensional hyperbolic equations with degenerate variable coefficients. The
two classes of SV methods are constructed by letting a piecewise -th order
( is an arbitrary integer) polynomial function satisfy the local
conservation law in each {\it control volume} obtained by dividing the interval
element of the underlying mesh with Gauss-Legendre points (LSV) or Radaus
points (RSV). The -norm stability and optimal order convergence properties
for both methods are rigorously proved for general non-uniform meshes. The
superconvergence behaviors of the two SV schemes have been also investigated:
it is proved that under the norm, the SV flux function approximates the
exact flux with -th order and the SV solution approximates the exact
solution with -th order; some superconvergence behaviors at
certain special points and for element averages have been also discovered and
proved. Our theoretical findings are verified by several numerical experiments
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