scholarly journals Implementation of high-order, discontinuous Galerkin time stepping for fractional diffusion problems

2021 ◽  
Vol 62 ◽  
pp. 121-147
Author(s):  
William McLean
Keyword(s):  
Discontinuous Galerkin ◽  
Fractional Order ◽  
Numerical Experiments ◽  
Time Integration ◽  
Fractional Diffusion ◽  
Practical Implementation ◽  
Time Stepping ◽  
Dg Method ◽  
Carry Over ◽  
Diffusion Problems

The discontinuous Galerkin (DG) method provides a robust and flexible technique for the time integration of fractional diffusion problems. However, a practical implementation uses coefficients defined by integrals that are not easily evaluated. We describe specialized quadrature techniques that efficiently maintain the overall accuracy of the DG method. In addition, we observe in numerical experiments that known superconvergence properties of DG time stepping for classical diffusion problems carry over in a modified form to the fractional-order setting. doi: 10.1017/S1446181120000152

scholarly journals IMPLEMENTATION OF HIGH-ORDER, DISCONTINUOUS GALERKIN TIME STEPPING FOR FRACTIONAL DIFFUSION PROBLEMS

2020 ◽  
Vol 62 (2) ◽  
pp. 121-147
Author(s):  
WILLIAM MCLEAN
Keyword(s):  
Discontinuous Galerkin ◽  
Fractional Order ◽  
Numerical Experiments ◽  
Time Integration ◽  
Fractional Diffusion ◽  
High Order ◽  
Practical Implementation ◽  
Time Stepping ◽  
Carry Over ◽  
Diffusion Problems

AbstractThe discontinuous Galerkin (DG) method provides a robust and flexible technique for the time integration of fractional diffusion problems. However, a practical implementation uses coefficients defined by integrals that are not easily evaluated. We describe specialized quadrature techniques that efficiently maintain the overall accuracy of the DG method. In addition, we observe in numerical experiments that known superconvergence properties of DG time stepping for classical diffusion problems carry over in a modified form to the fractional-order setting.

scholarly journals Some Time Stepping Methods for Fractional Diffusion Problems with Nonsmooth Data

2018 ◽  
Vol 18 (1) ◽  
pp. 129-146 ◽  
Author(s):  
Yan Yang ◽  
Yubin Yan ◽  
Neville J. Ford
Keyword(s):  
Initial Data ◽  
Convergence Rate ◽  
Homogeneous Problem ◽  
Fractional Diffusion ◽  
Nonsmooth Data ◽  
Time Stepping ◽  
Time Discretisation ◽  
Nonsmooth Initial Data ◽  
Densely Defined ◽  
Diffusion Problems

AbstractWe consider error estimates for some time stepping methods for solving fractional diffusion problems with nonsmooth data in both homogeneous and inhomogeneous cases. McLean and Mustapha [18] established an {O(k)} convergence rate for the piecewise constant discontinuous Galerkin method with nonsmooth initial data for the homogeneous problem when the linear operator A is assumed to be self-adjoint, positive semidefinite and densely defined in a suitable Hilbert space, where k denotes the time step size. In this paper, we approximate the Riemann–Liouville fractional derivative by Diethelm’s method (or L1 scheme) and obtain the same time discretisation scheme as in McLean and Mustapha [18]. We first prove that this scheme has also convergence rate {O(k)} with nonsmooth initial data for the homogeneous problem when A is a closed, densely defined linear operator satisfying some certain resolvent estimates. We then introduce a new time discretisation scheme for the homogeneous problem based on the convolution quadrature and prove that the convergence rate of this new scheme is {O(k^{1+\alpha})}, {0<\alpha<1}, with the nonsmooth initial data. Using this new time discretisation scheme for the homogeneous problem, we define a time stepping method for the inhomogeneous problem and prove that the convergence rate of this method is {O(k^{1+\alpha})}, {0<\alpha<1}, with the nonsmooth data. Numerical examples are given to show that the numerical results are consistent with the theoretical results.

scholarly journals DG approach to large bending plate deformations with isometry constraint

2021 ◽  
pp. 1-43
Author(s):  
Andrea Bonito ◽  
Ricardo H. Nochetto ◽  
Dimitrios Ntogkas
Keyword(s):  
Discontinuous Galerkin ◽  
Minimization Problem ◽  
Numerical Experiments ◽  
Gradient Flow ◽  
Kirchhoff Plate ◽  
Geometrically Nonlinear ◽  
Plate Model ◽  
Global Minimizers ◽  
Dg Method ◽  
Kirchhoff Plate Model

We propose a new discontinuous Galerkin (dG) method for a geometrically nonlinear Kirchhoff plate model for large isometric bending deformations. The minimization problem is nonconvex due to the isometry constraint. We present a practical discrete gradient flow that decreases the energy and computes discrete minimizers that satisfy a prescribed discrete isometry defect. We prove [Formula: see text]-convergence of the discrete energies and discrete global minimizers. We document the flexibility and accuracy of the dG method with several numerical experiments.

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