In this paper, the Navier-Stokes-Darcy equations with the free interface are considered, which model the movement of the sea and the sand in the seafloor or the filtration of blood through the arterial wall. The global well-posedness of the solution perturbed around the constant steady state is obtained and then the almost exponential decay to the constant stationary state is gained. Finally, we present an efficient explicit discrete scheme based on finite-volume method for the free interface system and provide the numerical tests to illustrate the consistency with our analysis result.

For the heat equation on a rectangle and nonzero Dirichlet boundary conditions, we consider an ADI orthogonal spline collocation method without perturbation terms, to specify boundary values of intermediate solutions at half time levels on the vertical sides of the rectangle. We show that, at each time level, the method has optimal convergence rate in the $L^2$ norm in space. Numerical results for splines of orders 4, 5, 6 confirm our theoretical convergence rates and demonstrate suboptimal convergence rates in the $H^1$ norm. We also demonstrate numerically that the scheme without the perturbation terms is applicable to variable coefficient problems yielding the same convergence rates obtained for the heat equation.

The design of numerical approaches for the molecular beam epitaxy models has always been a hot issue in numerical analysis, in which one of the main challenges for algorithm design is how to establish a high-order time-accurate numerical method with unconditional energy stability. The numerical method developed in this paper is based on the “stabilized-Invariant Energy Quadratization” (S-IEQ) approach. Its novelty is that by adding a very simple linear stabilization term, the difficulty that the original energy potential for the no-slope selection case is not bounded from below can be easily overcome. Then by using the standard format of the IEQ method, we can easily obtain a linear, unconditionally energy stable, and second-order time accurate scheme for solving the system. We further implement various numerical examples to demonstrate the stability and accuracy of the proposed scheme.

We introduce a flux recovery scheme for an enriched finite element method applied to an interface diffusion equation with absorption. The method is a variant of the finite element method introduced by Wang $et$ $al$. in [20]. The recovery is done at nodes first and then extended to the whole domain by interpolation. In the case of piecewise constant diffusion coefficient, we show that the nodes of the finite elements are superconvergence points for both the primary variable $p$ and its flux $u$. In particular, in the absence of the absorption term zero error is achieved at the nodes and interface point in the approximation of $u$ and $p$. In the general case, pressure error at the nodes and interface point is second order. Numerical results are provided to confirm the theory.

We propose a new, efficient, nonlinear iteration for solving the steady incompressible MHD equations. The method consists of a careful combination of an incremental Picard iteration, Yosida splitting, and a grad-div stabilized finite element discretization. At each iteration, the Schur complement remains the same, is SPD, and can be easily and effectively preconditioned with the pressure mass matrix. Furthermore, this method decouples the block Schur complement into 2 simple Stokes Schur complement. We show that the iteration converges linearly to the discrete MHD system solution, both analytically and numerically. Several numerical tests are given which reveal very good convergence properties, and excellent results on a benchmark problem.

We consider extrapolation of the Arnoldi algorithm to accelerate computation of the dominant eigenvalue/eigenvector pair. The basic algorithm uses sequences of Krylov vectors to form a small eigenproblem which is solved exactly. The two dominant eigenvectors output from consecutive Arnoldi steps are then recombined to form an extrapolated iterate, and this accelerated iterate is used to restart the next Arnoldi process. We present numerical results testing the algorithm on a variety of cases and find on most examples it substantially improves the performance of restarted Arnoldi. The extrapolation is a simple post-processing step which has minimal computational cost.