An official website of the United States government
The classical continuous finite element method with Lagrangian Q^k basis reduces to a finite difference scheme when all the integrals are replaced by the (𝑘+1)×(𝑘+1) Gauss–Lobatto quadrature. We prove that this finite difference scheme is (𝑘+2)-th order accurate in the discrete 2-norm for an elliptic equation with Dirichlet boundary conditions, which is a superconvergence result of function values. We also give a convenient implementation for the case 𝑘=2, which is a simple fourth order accurate elliptic solver on a rectangular domain.
more »
« less
Analysis of optimal superconvergence of an ultraweak-local discontinuous Galerkin method for a time dependent fourth-order equation
In this paper, we study superconvergence properties of the ultraweak-local discontinuous Galerkin (UWLDG) method in Tao et al. [To appear in Math. Comput. DOI: https://doi.org/10.1090/mcom/3562 (2020).] for an one-dimensional linear fourth-order equation. With special initial discretizations, we prove the numerical solution of the semi-discrete UWLDG scheme superconverges to a special projection of the exact solution. The order of this superconvergence is proved to be k + min(3, k ) when piecewise ℙ k polynomials with k ≥ 2 are used. We also prove a 2 k -th order superconvergence rate for the cell averages and for the function values and derivatives of the UWLDG approximation at cell boundaries. Moreover, we prove superconvergence of ( k + 2)-th and ( k + 1)-th order of the function values and the first order derivatives of the UWLDG solution at a class of special quadrature points, respectively. Our proof is valid for arbitrary non-uniform regular meshes and for arbitrary k ≥ 2. Numerical experiments verify that all theoretical findings are sharp.
more »
« less
- Award ID(s):
- 1719410
- PAR ID:
- 10226101
- Date Published:
- Journal Name:
- ESAIM: Mathematical Modelling and Numerical Analysis
- Volume:
- 54
- Issue:
- 6
- ISSN:
- 0764-583X
- Page Range / eLocation ID:
- 1797 to 1820
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Abstract In this paper, we study the superconvergence of the semi-discrete discontinuous Galerkin (DG) method for linear hyperbolic equations in one spatial dimension. The asymptotic errors in cell averages, downwind point values, and the postprocessed solution are derived for the initial discretization by Gaussian projection (for periodic boundary condition) or Cao projection Cao et al. (SIAM J. Numer. Anal.5, 2555–2573 (2014)) (for Dirichlet boundary condition). We proved that the error constant in the superconvergence of order$$2k+1$$ for DG methods based on upwind-biased fluxes depends on the parity of the orderk. The asymptotic errors are demonstrated by various numerical experiments for scalar and vector hyperbolic equations.more » « less
-
Abstract In this article, we study the spectral volume (SV) methods for scalar hyperbolic conservation laws with a class of subdivision points under the Petrov–Galerkin framework. Due to the strong connection between the DG method and the SV method with the appropriate choice of the subdivision points, it is natural to analyze the SV method in the Galerkin form and derive the analogous theoretical results as in the DG method. This article considers a class of SV methods, whose subdivision points are the zeros of a specific polynomial with a parameter in it. Properties of the piecewise constant functions under this subdivision, including the orthogonality between the trial solution space and test function space, are provided. With the aid of these properties, we are able to derive the energy stability, optimal a priori error estimates of SV methods with arbitrary high order accuracy. We also study the superconvergence of the numerical solution with the correction function technique, and show the order of superconvergence would be different with different choices of the subdivision points. In the numerical experiments, by choosing different parameters in the SV method, the theoretical findings are confirmed by the numerical results.more » « less
-
In this paper, we study the optimal error estimates of the classical discontinuous Galerkin method for time-dependent 2-D hyperbolic equations using P k elements on uniform Cartesian meshes, and prove that the error in the L 2 norm achieves optimal ( k + 1)th order convergence when upwind fluxes are used. For the linear constant coefficient case, the results hold true for arbitrary piecewise polynomials of degree k ≥ 0. For variable coefficient and nonlinear cases, we give the proof for piecewise polynomials of degree k = 0, 1, 2, 3 and k = 2, 3, respectively, under the condition that the wind direction does not change. The theoretical results are verified by numerical examples.more » « less
-
Abstract We construct special cycles on the moduli stack of hermitian shtukas. We prove an identity between (1) the$$r^{\mathrm{th}}$$ central derivative of non-singular Fourier coefficients of a normalized Siegel–Eisenstein series, and (2) the degree of special cycles of “virtual dimension 0” on the moduli stack of hermitian shtukas with$$r$$ legs. This may be viewed as a function-field analogue of the Kudla-Rapoport Conjecture, that has the additional feature of encompassing all higher derivatives of the Eisenstein series.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1051/m2an/2020023
