This paper develops a posteriori error estimates of residual type for conforming and mixed finite element approximations of the fourth order Cahn-Hilliard equation $u_t+∆(ε∆u−ε^{-1}f(u))=0$. It is shown that the a posteriori error bounds depends on $ε^{-1}$ only in some low polynomial order, instead of exponential order. Using these a posteriori error estimates, we construct an adaptive algorithm for computing the solution of the Cahn-Hilliard equation and its sharp interface limit, the Hele-Shaw flow. Numerical experiments are presented to show the robustness and effectiveness of the new error estimators and the proposed adaptive algorithm.

The convergence analysis on the general iterative methods for the symmetric and positive semidefinite problems is presented in this paper. First, formulated are refined necessary and sufficient conditions for the energy norm convergence for iterative methods. Some illustrative examples for the conditions are also provided. The sharp convergence rate identity for the Gauss-Seidel method for the semidefinite system is obtained relying only on the pure matrix manipulations which guides us to obtain the convergence rate identity for the general successive subspace correction methods. The convergence rate identity for the successive subspace correction methods is obtained under the new conditions that the local correction schemes possess the local energy norm convergence. A convergence rate estimate is then derived in terms of the exact subspace solvers and the parameters that appear in the conditions. The uniform convergence of multigrid method for a model problem is proved by the convergence rate identity. The work can be regraded as unified and simplified analysis on the convergence of iteration methods for semidefinite problems [8,9].

In this paper, we introduce a mixed finite element method on a staggered mesh for the numerical solution of the steady state Navier-Stokes equations in which the two components of the velocity and the pressure are defined on three different meshes. This method is a conforming quadrilateral $Q_1\times Q_1-P_0$ element approximation for the Navier-Stokes equations. First-order error estimates are obtained for both the velocity and the pressure. Numerical examples are presented to illustrate the effectiveness of the proposed method.

The main purpose of this work is to provide a novel numerical approach for the Volterra integral equations based on a spectral approach. A Legendre-collocation method is proposed to solve the Volterra integral equations of the second kind. We provide a rigorous error analysis for the proposed method, which indicates that the numerical errors decay exponentially provided that the kernel function and the source function are sufficiently smooth. Numerical results confirm the theoretical prediction of the exponential rate of convergence. The result in this work seems to be the first successful spectral approach (with theoretical justification) for the Volterra type equations.

Implicit-explicit (IMEX) linear multistep methods are popular techniques for solving partial differential equations (PDEs) with terms of different types. While fixed time-step versions of such schemes have been developed and studied, implicit-explicit schemes also naturally arise in general situations where the temporal smoothness of the solution changes. In this paper we consider easily implementable variable step-size implicit-explicit (VSIMEX) linear multistep methods for time-dependent PDEs. Families of order-$p$, $p$-step VSIMEX schemes are constructed and analyzed, where $p$ ranges from 1 to 4. The corresponding schemes are simple to implement and have the property that they reduce to the classical IMEX schemes whenever constant time step-sizes are imposed. The methods are validated on the Burgers' equation. These results demonstrate that by varying the time step-size, VSIMEX methods can outperform their fixed time step counterparts while still maintaining good numerical behavior.

Digital inpainting is a fundamental problem in image processing and many variational models for this problem have appeared recently in the literature. Among them are the very successfully Total Variation (TV) model [11] designed for local inpainting and its improved version for large scale inpainting: the Curvature-Driven Diffusion (CDD) model [10]. For the above two models, their associated Euler Lagrange equations are highly nonlinear partial differential equations. For the TV model there exists a relatively fast and easy to implement fixed point method, so adapting the multigrid method of [24] to here is immediate. For the CDD model however, so far only the well known but usually very slow explicit time marching method has been reported and we explain why the implementation of a fixed point method for the CDD model is not straightforward. Consequently, the multigrid method as in [Savage and Chen, Int. J. Comput. Math., 82 (2005), pp. 1001-1015] will not work here. This fact represents a strong limitation to the range of applications of this model since usually fast solutions are expected. In this paper, we introduce a modification designed to enable a fixed point method to work and to preserve the features of the original CDD model. As a result, a fast and efficient multigrid method is developed for the modified model. Numerical experiments are presented to show the very good performance of the fast algorithm.

The inverse black body radiation problem, which is to reconstruct the area temperature distribution from the measurement of power spectrum distribution, is a well-known ill-posed problem. In this paper, a variational expectation-maximization (EM) method is developed and its convergence is studied. Numerical experiments demonstrate that the variational EM method is more efficient and accurate than the traditional methods, including the Tikhonov regularization method, the Landweber method and the conjugate gradient method.