We consider the convergence theory of adaptive multigrid methods for second-order elliptic problems and Maxwell's equations. The multigrid algorithm only performs pointwise Gauss-Seidel relaxations on new degrees of freedom and their "immediate" neighbors. In the context of lowest order conforming finite element approximations, we present a unified proof for the convergence of adaptive multigrid V-cycle algorithms. The theory applies to any hierarchical tetrahedral meshes with uniformly bounded shape-regularity measures. The convergence rates for both problems are uniform with respect to the number of mesh levels and the number of degrees of freedom. We demonstrate our convergence theory by two numerical experiments.

As the generalization of the integer order partial differential equations (PDE), the fractional order PDEs are drawing more and more attention for their applications in fluid flow, finance and other areas. This paper presents high-order accurate Runge-Kutta local discontinuous Galerkin (DG) methods for one- and two-dimensional fractional diffusion equations containing derivatives of fractional order in space. The Caputo derivative is chosen as the representation of spatial derivative, because it may represent the fractional derivative  by an integral operator. Some numerical examples show that the convergence orders of the proposed local $P^k$-DG methods are $O(h^{k+1})$ both in one and two dimensions, where $P^k$ denotes the space of the real-valued polynomials with degree at most $k$.

In this paper, we study robust iterative solvers for finite element systems resulting in approximation of steady-state Richards' equation in porous media with highly heterogeneous conductivity fields. It is known that in such cases the contrast, ratio between the highest and lowest values of the conductivity, can adversely affect the performance of the preconditioners and, consequently, a design of robust preconditioners is important for many practical applications. The proposed iterative solvers consist of two kinds of iterations, outer and inner iterations. Outer iterations are designed to handle nonlinearities by linearizing the equation around the previous solution state. As a result of the linearization, a large-scale linear system needs to be solved. This linear system is solved iteratively (called inner iterations), and since it can have large variations in the coefficients, a robust preconditioner is needed. First, we show that under some assumptions the number of outer iterations is independent of the contrast. Second, based on the recently developed iterative methods, we construct a class of preconditioners that yields convergence rate that is independent of the contrast. Thus, the proposed iterative solvers are optimal with respect to the large variation in the physical parameters. Since the same preconditioner can be reused in every outer iteration, this provides an additional computational savings in the overall solution process. Numerical tests are presented to confirm the theoretical results.

Node importance or centrality evaluation is an important methodology for network analysis. In this paper, we are interested in the study of objects appearing in several networks. Such common objects are important in network-network interactions via object-object interactions. The main contribution of this paper is to model multiple networks where there are some common objects in a multivariate Markov chain framework, and to develop a method for solving common and non-common objects' stationary probability distributions in the networks. The stationary probability distributions can be used to evaluate the importance of common and non-common objects via network-network interactions. Our experimental results based on examples of co-authorship of researchers in different conferences and paper citations in different categories have shown that the proposed model can provide useful information for researcher-researcher interactions in networks of different conferences and for paper-paper interactions in networks of different categories.

For weighted approximation in $L_p$ -norm, we determine strongly asymptotic orders for the average errors of both function approximation and derivative approximation by the Bernstein operators sequence on the $r$-fold integrated Wiener space.

In this paper, we investigate the superconvergence property and the $L^{\infty}$-error estimates of mixed finite element methods for a semilinear elliptic control problem with an integral constraint. The state and co-state are approximated by the order one Raviart-Thomas mixed finite element space and the control variable is approximated by piecewise constant functions or piecewise linear functions. We derive some superconvergence results for the control variable and the state variables when the control is approximated by piecewise constant functions. Moreover, we derive $L^{\infty}$-error estimates for both the control variable and the state variables when the control is discretized by piecewise linear functions. Finally, some numerical examples are given to demonstrate the theoretical results.  

With the aid of index functions, we re-derive the ML($n$)BiCGStab algorithm in [Yeung and Chan, SIAM J. Sci. Comput., 21 (1999), pp. 1263-1290] systematically. There are $n$ ways to define the ML($n$)BiCGStab residual vector. Each definition leads to a different ML($n$)BiCGStab algorithm. We demonstrate this by presenting a second algorithm which requires less storage. In theory, this second algorithm serves as a bridge that connects the Lanczos-based BiCGStab and the Arnoldi-based FOM while ML($n$)BiCG is a bridge connecting BiCG and FOM. We also analyze the breakdown situation from the probabilistic point of view and summarize some useful properties of ML($n$)BiCGStab. Implementation issues are also addressed.

In this paper, we introduce the dissipative spectral methods (DSM) for the first order linear hyperbolic equations in one dimension. Specifically, we consider the Fourier DSM for periodic problems and the Legendre DSM for equations with the Dirichlet boundary condition. The error estimates of the methods are shown to be quasi-optimal for variable-coefficients equations. Numerical results are given to verify high accuracy of the DSM and to compare the proposed schemes with some high performance methods, showing some superiority in long-term integration for the periodic case and in dealing with limited smoothness near or at the boundary for the Dirichlet case.