Time-stepping discontinuous Galerkin approximation of optimal control problem governed by time fractional diffusion equation

In this paper, a piecewise constant time-stepping discontinuous Galerkin method combined with a piecewise linear finite element method is applied to solve control constrained optimal control problem governed by time fractional diffusion equation. The control variable is approximated by variational discretization approach. The discrete first-order optimality condition is derived based on the first discretize then optimize approach. We demonstrate the commutativity of discretization and optimization for the time-stepping discontinuous Galerkin discretization. Since the state variable and the adjoint state variable in general have weak singularity near t =?0and t = T, a time adaptive algorithm is developed based on step doubling technique, which can be used to guide the time mesh refinement. Numerical examples are given to illustrate the theoretical findings.     

Piecewise-linear,discontinuous Galerkin method for a fractional diffusion equation

We use a piecewise-linear, discontinuous Galerkin method for the time discretization of a fractional diffusion equation involving a parameter in the range − 1 < α < 0. Our analysis shows that, for a time interval (0,T) and a spatial domain Ω, the error in L_￥((0,T);L₂(W))L_\infty\bigr((0,T);L_2(\Omega)\bigr) is of order k ^2 + α , where k denotes the maximum time step. Since derivatives of the solution may be singular at t = 0, our result requires the use of non-uniform time steps. In the limiting case α = 0 we recover the known O(k ²) convergence for the classical diffusion (heat) equation. We also consider a fully-discrete scheme that employs standard (continuous) piecewise-linear finite elements in space, and show that the additional error is of order h ²log(1/k). Numerical experiments indicate that our O(k ^2 + α ) error bound is pessimistic. In practice, we observe O(k ²) convergence even for α close to − 1.     

Local discontinuous Galerkin methods with explicit-implicit-null time discretizations for solving nonlinear diffusion problems

In this paper,we discuss the local discontinuous Galerkin methods coupled with two specific explicitimplicit-null time discretizations for solving one-dimensional nonlinear diffusion problems Ut=(a(U)Ux)x.The basic idea is to add and subtract two equal terms a0 Uxx the right-hand side of the partial differential equation,then to treat the term a0 Uxx implicitly and the other terms(a(U)Ux)x-a0 Uxx explicitly.We give stability analysis for the method on a simplified model by the aid of energy analysis,which gives a guidance for the choice of a0,i.e.,a0≥max{a(u)}/2 to ensure the unconditional stability of the first order and second order schemes.The optimal error estimate is also derived for the simplified model,and numerical experiments are given to demonstrate the stability,accuracy and performance of the schemes for nonlinear diffusion equations.     

An analysis of discontinuous Galerkin methods for elliptic problems

We study global and local behaviors for three kinds of discontinuous Galerkin schemes for elliptic equations of second order. We particularly investigate several a posteriori error estimations for the discontinuous Galerkin schemes. These theoretical results are applied to develop local/parallel and adaptive finite element methods, based on the discontinuous Galerkin methods. Dedicated to Dr. Charles A. Micchelli on the occasion of his 60th birthday with friendship and esteem Mathematics subject classifications (2000) 65N12, 65N15, 65N30. Aihui Zhou: Subsidized by the Special Funds for Major State Basic Research Projects, and also partially supported by National Science Foundation of China. Reinhold Schneider: Supported in part by DFG Sonderforschungsbereich SFB 393. Yuesheng Xu: Correspondence author. Supported in part by the US National Science Foundation under grants DMS-9973427 and CCR-0312113, by Natural Science Foundation of China under grant 10371122 and by the Chinese Academy of Sciences under program “Hundreds Distinguished Young Chinese Scientists”.     

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 μ ∈ (0, 1) with variable coefficients. For the spatial discretization, we apply the standard continuous Galerkin method of total degree ≤ 1 on each spatial mesh elements. Well-posedness of the fully discrete scheme and error analysis will be shown. For a time interval (0, T) and a spatial domain Ω, our analysis suggest that the error in \(L^{2}\left ((0,T),L^{2}({\Omega })\right )\)-norm is \(O(k^{2-\frac {\mu }{2}}+h^{2})\) (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(k² + h²) error bound in the stronger \(L^{\infty }\left ((0,T),L^{2}({\Omega })\right )\)-norm. Variable time steps are used to compensate the singularity of the continuous solution near t = 0.     

BDDC methods for discontinuous Galerkin discretization of elliptic problems

A discontinuous Galerkin (DG) discretization of Dirichlet problem for second-order elliptic equations with discontinuous coefficients in 2-D is considered. For this discretization, balancing domain decomposition with constraints (BDDC) algorithms are designed and analyzed as an additive Schwarz method (ASM). The coarse and local problems are defined using special partitions of unity and edge constraints. Under certain assumptions on the coefficients and the mesh sizes across ∂Ω_i, where the Ω_i are disjoint subregions of the original region Ω, a condition number estimate C(1+max_ilog(H_i/h_i))² is established with C independent of h_i, H_i and the jumps of the coefficients. The algorithms are well suited for parallel computations and can be straightforwardly extended to the 3-D problems. Results of numerical tests are included which confirm the theoretical results and the necessity of the imposed assumptions.     

Multiscale method based on discontinuous Galerkin methods for homogenization problems

We propose a multiscale method for elliptic problems with highly oscillating coefficients based on a coupling of macro and micro methods in the framework of the heterogeneous multiscale method. The macro method, defined on a macroscopic triangulation, aims at recovering the effective (homogenized) solution of an unknown macro model. The unspecified data of this model are computed by micro methods on sampling domains during the macro assembly process. In this Note, we show how to construct such a coupling with a discontinuous macro finite element space. We show that the flux information needed in this formulation in order to impose weak interelement continuity can be recovered from the known micro calculations on the sampling domains. A fully discrete analysis is presented. To cite this article: A. Abdulle, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Robust smoothers for high-order discontinuous Galerkin discretizations of advection–diffusion problems

The multigrid method for discontinuous Galerkin (DG) discretizations of advection–diffusion problems is presented. It is based on a block Gauss–Seidel smoother with downwind ordering honoring the advection operator. The cell matrices of the DG scheme are inverted in this smoother in order to obtain robustness for higher order elements. Employing a set of experiments, we show that this technique actually yields an efficient preconditioner and that both ingredients, downwind ordering and blocking of cell matrices are crucial for robustness.     

A posteriori error estimates for discontinuous Galerkin methods of obstacle problems

We present a posteriori error analysis of discontinuous Galerkin methods for solving the obstacle problem, which is a representative elliptic variational inequality of the first kind. We derive reliable error estimators of the residual type. Efficiency of the estimators is theoretically explored and numerically confirmed.     

Discontinuous Galerkin isogeometric analysis for elliptic problems with discontinuous diffusion coefficients on surfaces

Numerical Algorithms - This paper is concerned with using discontinuous Galerkin isogeometric analysis (dG-IGA) as a numerical treatment of diffusion problems on orientable surfaces ${Omega }...     

Abstract theory of hybridizable discontinuous Galerkin methods for second-order quasilinear elliptic problems

An abstract theory for discretizations of second-order quasilinear elliptic problems based on the mixed-hybrid discontinuous Galerkin method. Discrete schemes are formulated in terms of approximations of the solution to the problem, its gradient, flux, and the trace of the solution on the interelement boundaries. Stability and optimal error estimates are obtained under minimal assumptions on the approximating space. It is shown that the schemes admit an efficient numerical implementation.     

One-step Taylor–Galerkin methods for convection–diffusion problems

Third and fourth order Taylor–Galerkin schemes have shown to be efficient finite element schemes for the numerical simulation of time-dependent convective transport problems. By contrast, the application of higher-order Taylor–Galerkin schemes to mixed problems describing transient transport by both convection and diffusion appears to be much more difficult. In this paper we develop two new Taylor–Galerkin schemes maintaining the accuracy properties and improving the stability restrictions in convection–diffusion. We also present an efficient algorithm for solving the resulting system of the finite element method. Finally we present two numerical simulations that confirm the properties of the methods.     

The discontinuous Galerkin method for fractional degenerate convection-diffusion equations

We propose and study discontinuous Galerkin methods for strongly degenerate convection-diffusion equations perturbed by a fractional diffusion (Lévy) operator. We prove various stability estimates along with convergence results toward properly defined (entropy) solutions of linear and nonlinear equations. Finally, the qualitative behavior of solutions of such equations are illustrated through numerical experiments.     

Pointwise error estimates for discontinuous Galerkin methods with lifting operators for elliptic problems

In this article, we prove some weighted pointwise estimates for three discontinuous Galerkin methods with lifting operators appearing in their corresponding bilinear forms. We consider a Dirichlet problem with a general second-order elliptic operator.

     

对流扩散方程的时空间断有限元方法

研究对流扩散方程的时空间断Galerkin有限元方法,该方法采用时,空两个变量都允许间断的基函数,更适用于移动网格,自适应算法以及并行计算.本文利用拉格朗日欧拉方法,采用F.Brezzi数值流通量,给出对流扩散方程的间断时空有限元离散格式,并证明格式的相容性,强制性,稳定性,解的存在唯一性,以及总体误差估计.     

Rational Krylov methods for fractional diffusion problems on graphs

BIT Numerical Mathematics - In this paper we propose a method to compute the solution to the fractional diffusion equation on directed networks, which can be expressed in terms of the graph...