We consider the explicit numerical approximations of stochastic differential equations (SDEs) driven by Brownian process and Poisson jump. It is well known that under non-global Lipschitz condition, Euler Explicit method fails to converge strongly to the exact solution of such SDEs without jumps, while implicit Euler method converges but requires much computational efforts. We investigate the strong convergence, the linear and nonlinear exponential stabilities of tamed Euler and semi-tamed methods for stochastic differential equation driven by Brownian process and Poisson jumps, both in compensated and non compensated forms. We prove that under non-global Lipschitz condition and superlinearly growing drift term, these schemes converge strongly with the standard one-half order. Numerical simulations to sustain the theoretical results are provided.

In this paper we assess the parallel efficiency issues for simulating single phase subsurface flow in porous media, where the permeability tensor contains anisotropy rotated with certain angles or severe discontinuity. Space variables are discretized using multi-points flux approximations and the pressure equations are solved by aggregation-based algebraic multigrid method. The involved issues include the domain decomposition of space discretization and coarsening, smoothing, the coarsest grid solving of multigrid solving steps. Numerical experiments exhibit that the convergence of the multigrid algorithm suffers from the parallel implementation. The linear system at the coarsest grid is solved and by various iterative methods and the experimental results show that the parallel efficiency is less attenuated when sparse approximate inverse preconditioning conjugate gradient is used.

Many important physical problems, such as fluid structure interaction or conjugate heat transfer, require numerical methods that compute boundary derivatives or fluxes to high accuracy. This paper proposes a novel approach to calculating accurate approximations of boundary derivatives of elliptic problems. We describe a new continuous finite element method based on $p$-refinement of cells adjacent to the boundary that increases the local degree of the approximation. We prove that the order of the approximation on the $p$-refined cells is, in 1D, determined by the rate of convergence at the mesh vertex connecting the higher and lower degree cells and that this approach can be extended, in a restricted setting, to 2D problems. The proven convergence rates are numerically verified by a series of experiments in both 1D and 2D. Finally, we demonstrate, with additional numerical experiments, that the $p$-refinement method works in more general geometries.

In this paper, a modified nonstandard finite difference method for the two-dimensional Riesz space fractional reaction-diffusion equations is developed. The space fractional derivative is discretized by the shifted Grünwald-Letnikov method and the nonlinear reaction term is approximated by Taylor formula instead of Micken's. Multigrid method is introduced to reduce the computation time of the traditional Gauss-Seidal method. The stability and convergence of the nonstandard implicit difference scheme are strictly proved. The method is extended to simulate the fractional FitzHugh-Nagumo model. Numerical results are provided to verify the theoretical analysis.

In this paper, the distributed Lagrange multiplier-fictitious domain (DLM/FD) finite element method is studied for a type of steady state Stokes interface problems with jump coefficients, and its well-posedness, stability and optimal convergence properties are analyzed by proving an $inf$-$sup$ condition for a nested saddle-point problem that is induced by both Stokes equations and DLM/FD method in regard to Stokes variables (velocity and pressure) and Lagrange multipliers. Numerical experiments validate the obtained convergence theorem of DLM/FD finite element method for Stokes interface problems with respect to different jump ratios.

In this paper, we analyze a special immersed finite volume method that is different from the classic immersed finite volume method by choosing special control volumes near the interface. Using the elementwise stiffness matrix analysis technique and the $H^1$-norm-equivalence between the immersed finite element space and the standard finite element space, we prove that the special finite volume method is uniformly stable independent of the location of the interface. Based on the stability, we show that our scheme converges with the optimal order $\mathcal{O}$($h$) in the $H^1$ space and the order $\mathcal{O}$($h^{3/2}$) in the $L^2$ space. Numerically, we observe that our method converges with the optimal convergence rate $\mathcal{O}$($h$) under the $H^1$ norm and with the the optimal convergence rate $\mathcal{O}$($h^2$) under the $L^2$ norm all the way even with very small mesh size $h$, while the classic immersed finite element method is not able to maintain the optimal convergence rates (with diminished rate up to $\mathcal{O}$($h^{0.82}$) for the $H^1$ norm  error  and  diminished  rate  up to $\mathcal{O}$($h^{1.1}$) for $L^2$-norm error), when $h$ is getting small, as illustrated in Tables 4 and 5 of [35].