Wenqiang Feng

Non-iterative Domain Decomposition Methods For a Non-stationary Stokes-Darcy Model with Beavers-Joseph Interface Condition

Abstract In order to solve a non-stationary Stokes–Darcy model with Beavers–Joseph interface condition, two non-iterative domain decomposition methods are proposed. At each time step, results from previous time steps are utilized to approximate the information on the interface and decouple the two physics. Both of the two methods are parallel. Numerical results suggest that the first method has accuracy order O ( h 3 + Δ t ) . In order to improve the accuracy and efficiency, a three-step backward differentiation is used in the second method to achieve an accuracy order O ( h 3 + Δ t 3 ) , which is illustrated by a numerical example.

Preconditioned steepest descent methods for some nonlinear elliptic equations involving p-Laplacian terms

Abstract We describe and analyze preconditioned steepest descent (PSD) solvers for fourth and sixth-order nonlinear elliptic equations that include p-Laplacian terms on periodic domains in 2 and 3 dimensions. The highest and lowest order terms of the equations are constant-coefficient, positive linear operators, which suggests a natural preconditioning strategy. Such nonlinear elliptic equations often arise from time discretization of parabolic equations that model various biological and physical phenomena, in particular, liquid crystals, thin film epitaxial growth and phase transformations. The analyses of the schemes involve the characterization of the strictly convex energies associated with the equations. We first give a general framework for PSD in Hilbert spaces. Based on certain reasonable assumptions of the linear pre-conditioner, a geometric convergence rate is shown for the nonlinear PSD iteration. We then apply the general theory to the fourth and sixth-order problems of interest, making use of Sobolev embedding and regularity results to confirm the appropriateness of our pre-conditioners for the regularized p-Lapacian problems. Our results include a sharper theoretical convergence result for p-Laplacian systems compared to what may be found in existing works. We demonstrate rigorously how to apply the theory in the finite dimensional setting using finite difference discretization methods. Numerical simulations for some important physical application problems – including thin film epitaxy with slope selection and the square phase field crystal model – are carried out to verify the efficiency of the scheme.

A Uniquely Solvable, Energy Stable Numerical Scheme for the Functionalized Cahn–Hilliard Equation and Its Convergence Analysis

We present and analyze a uniquely solvable and unconditionally energy stable numerical scheme for the Functionalized Cahn–Hilliard equation, including an analysis of convergence. One key difficulty associated with the energy stability is based on the fact that one nonlinear energy functional term in the expansion is neither convex nor concave. To overcome this subtle difficulty, we add two auxiliary terms to make the combined term convex, which in turns yields a convex–concave decomposition of the physical energy. As a result, both the unconditional unique solvability and the unconditional energy stability of the proposed numerical scheme are assured. In addition, a global in time

A mass-conservative adaptive FAS multigrid solver for cell-centered finite difference methods on block-structured, locally-cartesian grids

H_{\mathrm{per}}^2

An energy stable fourth order finite difference scheme for the Cahn–Hilliard equation

Hper2 stability of the numerical scheme is established at a theoretical level, which in turn ensures the full order convergence analysis of the scheme, which is the first such result in this field. To deal with an implicit 4-Laplacian term at each time step, we apply an efficient preconditioned steepest descent algorithm to solve the corresponding nonlinear systems in the finite difference set-up. A few numerical results are presented, which confirm the stability and accuracy of the proposed numerical scheme.

A Fourier pseudospectral method for the “good” Boussinesq equation with second‐order temporal accuracy

In this paper, we propose a new adaptive non-local means algorithm for image denoising based on the region via pixel region growing and merging. For a seed pixel, by employing the pixel region growing and merging technique to the search regions, it lets the search regions cluster under the similar gray and structure adaptively and merge the related neighborhoods. Experiment results demonstrate that this method is efficient and robust not only for Gaussian white noise, but also for salts and peppers noise. Compared with ONL means and NLR means, the proposed method can not only accelerate the computation, but also have a high quality.

Immersed finite element method for interface problems with algebraic multigrid solver

Abstract We present a mass-conservative full approximation storage (FAS) multigrid solver for cell-centered finite difference methods on block-structured, locally cartesian grids. The algorithm is essentially a standard adaptive FAS (AFAS) scheme, but with a simple modification that comes in the form of a mass-conservative correction to the coarse-level force. This correction is facilitated by the creation of a zombie variable, analogous to a ghost variable, but defined on the coarse grid and lying under the fine grid refinement patch. We show that a number of different types of fine-level ghost cell interpolation strategies could be used in our framework, including low-order linear interpolation. In our approach, the smoother, prolongation, and restriction operations need never be aware of the mass conservation conditions at the coarse-fine interface. To maintain global mass conservation, we need only modify the usual FAS algorithm by correcting the coarse-level force function at points adjacent to the coarse-fine interface. We demonstrate through simulations that the solver converges geometrically, at a rate that is h -independent, and we show the generality of the solver, applying it to several nonlinear, time-dependent, and multi-dimensional problems. In several tests, we show that second-order asymptotic ( h → 0 ) convergence is observed for the discretizations, provided that (1) at least linear interpolation of the ghost variables is employed, and (2) the mass conservation corrections are applied to the coarse-level force term.

A second‐order energy stable backward differentiation formula method for the epitaxial thin film equation with slope selection

Some new nonlinear weakly singular difference inequalities are discussed, which generalize some known weakly singular inequalities and can be used in the analysis of nonlinear Volterra-type difference equations with weakly singular kernel. An application to the upper bound of solutions of a nonlinear difference equation is also presented.