Domain Decomposition Methods for Space Fractional Partial Differential Equations

In this paper, a two-level additive Schwarz preconditioner is proposed for solving the algebraic systems resulting from the finite element approximations of space fractional partial differential equations (SFPDEs). It is shown that the condition number of the preconditioned system is bounded by C(1+H/\delta), where H is the maximum diameter of subdomains and \delta is the overlap size among the subdomains. Numerical results are given to support our theoretical findings.Comment: 19 pages, three figure

A discontinuous Petrov-Galerkin method for time-fractional diffusion equations

We propose and analyze a time-stepping discontinuous Petrov-Galerkin method combined with the continuous conforming finite element method in space for the numerical solution of time-fractional subdiffusion problems. We prove the existence, uniqueness and stability of approximate solutions, and derive error estimates. To achieve high order convergence rates from the time discretizations, the time mesh is graded appropriately near~$t=0$ to compensate the singular (temporal) behaviour of the exact solution near $t=0$ caused by the weakly singular kernel, but the spatial mesh is quasiuniform. In the $L_\infty((0,T);L_2(\Omega))$-norm ($(0,T)$ is the time domain and $\Omega$ is the spatial domain), for sufficiently graded time meshes, a global convergence of order $k^{m+\alpha/2}+h^{r+1}$ is shown, where $0<\alpha<1$ is the fractional exponent, $k$ is the maximum time step, $h$ is the maximum diameter of the spatial finite elements, and $m$ and $r$ are the degrees of approximate solutions in time and spatial variables, respectively. Numerical experiments indicate that our theoretical error bound is pessimistic. We observe that the error is of order ~$k^{m+1}+h^{r+1}$, that is, optimal in both variables.Comment: SIAM Journal on Numerical Analysis, 201

Multigrid Methods for Space Fractional Partial Differential Equations

We propose some multigrid methods for solving the algebraic systems resulting from finite element approximations of space fractional partial differential equations (SFPDEs). It is shown that our multigrid methods are optimal, which means the convergence rates of the methods are independent of the mesh size and mesh level. Moreover, our theoretical analysis and convergence results do not require regularity assumptions of the model problems. Numerical results are given to support our theoretical findings.Comment: 24 pages, 4 figure

Second-order LOD multigrid method for multidimensional Riesz fractional diffusion equation

We propose a locally one dimensional (LOD) finite difference method for multidimensional Riesz fractional diffusion equation with variable coefficients on a finite domain. The numerical method is second-order convergent in both space and time directions, and its unconditional stability is strictly proved. Comparing with the popular first-order finite difference method for fractional operator, the form of obtained matrix algebraic equation is changed from $(I-A)u^{k+1}=u^k+b^{k+1}$ to $(I-{\widetilde A})u^{k+1}=(I+{\widetilde B})u^k+{\tilde b}^{k+1/2}$; the three matrices $A$, ${\widetilde A}$ and ${\widetilde B}$ are all Toeplitz-like, i.e., they have completely same structure and the computational count for matrix vector multiplication is $\mathcal{O}(N {log} N)$; and the computational costs for solving the two matrix algebraic equations are almost the same. The LOD-multigrid method is used to solve the resulting matrix algebraic equation, and the computational count is $\mathcal{O}(N {log} N)$ and the required storage is $\mathcal{O}(N)$, where $N$ is the number of grid points. Finally, the extensive numerical experiments are performed to show the powerfulness of the second-order scheme and the LOD-multigrid method.Comment: 26 page

A discontinuous Galerkin method for time fractional diffusion equations with variable coefficients

We propose a piecewise-linear, time-stepping discontinuous Galerkin method to solve numerically a time fractional diffusion equation involving Caputo derivative of order $\mu\in (0,1)$ with variable coefficients. For the spatial discretization, we apply the standard piecewise linear continuous Galerkin method. Well-posedness of the fully discrete scheme and error analysis will be shown. For a time interval~$(0,T)$ and a spatial domain~$\Omega$, our analysis suggest that the error in $L^2\bigr((0,T),L^2(\Omega)\bigr)$-norm is of order $O(k^{2-\frac{\mu}{2}}+h^2)$ (that is, short by order $\frac{\mu}{2}$ from being optimal in time) where $k$ denotes the maximum time step, and $h$ is the maximum diameter of the elements of the (quasi-uniform) spatial mesh. However, our numerical experiments indicate optimal $O(k^{2}+h^2)$ error bound in the stronger $L^\infty\bigr((0,T),L^2(\Omega)\bigr)$-norm. Variable time steps are used to compensate the singularity of the continuous solution near $t=0$

An ADI Crank-Nicolson Orthogonal Spline Collocation Method for the Two-Dimensional Fractional Diffusion-Wave Equation

A new method is formulated and analyzed for the approximate solution of a two-dimensional time-fractional diffusion-wave equation. In this method, orthogonal spline collocation is used for the spatial discretization and, for the time-stepping, a novel alternating direction implicit (ADI) method based on the Crank-Nicolson method combined with the $L1$-approximation of the time Caputo derivative of order $\alpha\in(1,2)$. It is proved that this scheme is stable, and of optimal accuracy in various norms. Numerical experiments demonstrate the predicted global convergence rates and also superconvergence

Finite difference schemes for multi-term time-fractional mixed diffusion-wave equations

The multi-term time-fractional mixed diffusion-wave equations (TFMDWEs) are considered and the numerical method with its error analysis is presented in this paper. First, a $L2$ approximation is proved with first order accuracy to the Caputo fractional derivative of order $\beta \in (1,2).$ Then the approximation is applied to solve a one-dimensional TFMDWE and an implicit, compact difference scheme is constructed. Next, a rigorous error analysis of the proposed scheme is carried out by employing the energy method, and it is proved to be convergent with first order accuracy in time and fourth order in space, respectively. In addition, some results for the distributed order and two-dimensional extensions are also reported in this work. Subsequently, a practical fast solver with linearithmic complexity is provided with partial diagonalization technique. Finally, several numerical examples are given to demonstrate the accuracy and efficiency of proposed schemes.Comment: $L2$ approximation compact difference scheme distributed order fast solver convergenc

Time-stepping discontinuous Galerkin methods for fractional diffusion problems

Time-stepping $hp$-versions discontinuous Galerkin (DG) methods for the numerical solution of fractional subdiffusion problems of order $-\alpha$ with $-1<\alpha<0$ will be proposed and analyzed. Generic $hp$-version error estimates are derived after proving the stability of the approximate solution. For $h$-version DG approximations on appropriate graded meshes near$t=0$, we prove that the error is of order$O(k^{\max\{2,p\}+\frac{\alpha}{2}})$, where $k$ is the maximum time-step size and $p\ge 1$ is the uniform degree of the DG solution. For $hp$-version DG approximations, by employing geometrically refined time-steps and linearly increasing approximation orders, exponential rates of convergence in the number of temporal degrees of freedom are shown. Finally, some numerical tests are given

Unconditionally stable time splitting methods for the electrostatic analysis of solvated biomolecules

This work introduces novel unconditionally stable operator splitting methods for solving the time dependent nonlinear Poisson-Boltzmann (NPB) equation for the electrostatic analysis of solvated biomolecules. In a pseudo-transient continuation solution of the NPB equation, a long time integration is needed to reach the steady state. This calls for time stepping schemes that are stable and accurate for large time increments. The existing alternating direction implicit (ADI) methods for the NPB equation are known to be conditionally stable, although being fully implicit. To overcome this difficulty, we propose several new operator splitting schemes, in both multiplicative and additive styles, including locally one-dimensional (LOD) schemes and additive operator splitting (AOS) schemes. The proposed schemes become much more stable than the ADI methods, and some of them are indeed unconditionally stable in dealing with solvated proteins with source singularities and non-smooth solutions. Numerically, the orders of convergence in both space and time are found to be one. Nevertheless, the precision in calculating the electrostatic free energy is low, unless a small time increment is used. Further accuracy improvements are thus considered. After acceleration, the optimized LOD method can produce a reliable energy estimate by integrating for a small and fixed number of time steps. Since one only needs to solve a tridiagonal linear system in each independent one dimensional process, the overall computation is very efficient. The unconditionally stable LOD method scales linearly with respect to the number of atoms in the protein studies, and is over 20 times faster than the conditionally stable ADI methods

A second-order accurate implicit difference scheme for time fractional reaction-diffusion equation with variable coefficients and time drift term

An implicit finite difference scheme based on the $L2$-$1_{\sigma}$ formula is presented for a class of one-dimensional time fractional reaction-diffusion equations with variable coefficients and time drift term. The unconditional stability and convergence of this scheme are proved rigorously by the discrete energy method, and the optimal convergence order in the $L_2$-norm is $\mathcal{O}(\tau^2 + h^2)$ with time step $\tau$ and mesh size $h$. Then, the same measure is exploited to solve the two-dimensional case of this problem and a rigorous theoretical analysis of the stability and convergence is carried out. Several numerical simulations are provided to show the efficiency and accuracy of our proposed schemes and in the last numerical experiment of this work, three preconditioned iterative methods are employed for solving the linear system of the two-dimensional case.Comment: 27 pages, 5 figures, 5 table