Error estimates for the Crank-Nicolson in time, Finite Element in space (CNFE) discretization of the Navier-Stokes equations require application of the discrete Gronwall inequality, which leads to a time-step $(\Delta t)$ restriction. All known convergence analyses of the fully discrete CNFE with linear extrapolation rely on a similar  $\Delta t$-restriction. We show that CNFE with arbitrary-order extrapolation (denoted CNLE) is convergences optimally in the energy norm without any $\Delta t$-restriction. We prove that CNLE velocity and corresponding discrete time-derivative converge optimally in $l^∞(H^1)$ and $l^2(L^2)$ respectively under the mild condition  $\Delta t \leq Mh^{1/4}$ for any arbitrary $M > 0$ (e.g. independent of problem data, $h$, and  $\Delta t$) where $h > 0$ is the maximum mesh element diameter. Convergence in these higher order norms is needed to prove convergence estimates for pressure and the drag/lift force a fluid exerts on an obstacle. Our analysis exploits the extrapolated convective velocity to avoid any  $\Delta t$-restriction for convergence in the energy norm. However, the coupling between the extrapolated convecting velocity of usual CNLE and the a priori control of average velocities (characteristic of CN methods) rather than pointwise velocities (e.g. backward-Euler methods) in $l^2(H^1)$ is precisely the source of  $\Delta t$-restriction for convergence in higher-order norms.

In this paper we are interested in solving parabolic problems with a piecewise constant diffusion coefficient on structured Cartesian meshes. The aim of this paper is to investigate the applicability and convergence behavior of combining two non-conventional but innovative methods: the Laplace transformation method in the discretization of the time variable and the immerse finite element method (IFEM) in the discretization of the space variable. The Laplace transformation in time leads to a set of Helmholtz-like problems independent of each other, which can be solved in highly parallel. The employment of immerse finite elements (IFEs) makes it possible to use a structured mesh, such as a simple Cartesian mesh, for the discretization of the space variable even if the material interface (across which the diffusion coefficient is discontinuous) is non-trivial. Numerical examples presented indicate that the combination of these two methods can perform optimally from the point of view of the degrees of polynomial spaces employed in the IFE spaces.

In this article we aim to study finite volume approximations which approximate the solutions of convection-dominated problems possessing the so-called interior transition layers. The stiffness of such problems is due to a small parameter multiplied to the highest order derivative which introduces various transition layers at the boundaries and at the interior points where certain compatibility conditions do not meet. Here, we are interested in resolving interior transition layers at turning points. The proposed semi-analytic method features interior layer correctors which are obtained from singular perturbation analysis near the turning points. We demonstrate this method is efficient, stable and it shows 2nd-order convergence in the approximations.

In this article, we investigate the decay rate of the solutions of two water wave models with a nonlocal viscous term written in the KdV form $$u_t+u_x+\beta u_{xxx}+\frac{\sqrt v}{\sqrt \pi}\int^t_0\frac{u_t(s)}{\sqrt{t-s}}ds+uu_x=vu_{xx}$$ and $$u_t+u_x-\beta u_{txx}+\frac{\sqrt v}{\sqrt \pi}\int^t_0\frac{u_t(s)}{\sqrt{t-s}}ds+uu_x=vu_{xx}$$ in the BBM form. In order to realize this numerical study, a numerical scheme based on the $G^{\alpha}$-scheme is developed.

In this paper local discontinuous Galerkin method (LDG) was analyzed for solving 1-D convection-diffusion equations with a boundary layer near the outflow boundary. Local error estimates are established on quasi-uniform meshes with maximum mesh size $h$. On a subdomain with $O(h\ln(1/h))$ distance away from the outflow boundary, the $L^2$ error of the approximations to the solution and its derivative converges at the optimal rate $O(h^{k+1})$ when polynomials of degree at most $k$ are used. Numerical experiments illustrate that the rate of convergence is uniformly valid and sharp. The numerical comparison of the LDG method and the streamline-diffusion finite element method are also presented.

Efficient methods for solving linear algebraic equations are crucial to creating fast and accurate numerical simulations in many applications. In this paper, an algebraic multigrid (AMG) method, which combines the classical coarsening scheme by [19] with an energy-minimizing interpolation algorithm by [26], is employed and tested for subsurface water simulations. Based on numerical tests using real field data, our results suggest that the energy-minimizing algebraic multigrid method is efficient and, more importantly, very robust.

In this paper, we analyze the defect of the rotated 9-point finite difference scheme, and present an optimal 9-point finite difference scheme for the Helmholtz equation with perfectly matched layer (PML) in two dimensional domain. For this method, we give an error analysis for the numerical wavenumber’s approximation of the exact wavenumber. Moreover, based on minimizing the numerical dispersion, we propose global and refined choice strategies for choosing optimal parameters of the 9-point finite difference scheme. Numerical experiments are given to illustrate the improvement of the accuracy and the reduction of the numerical dispersion.

In this paper, we focus on an a posteriori residual-based error estimator for the $T/\Omega$ magnetodynamic harmonic formulation of the Maxwell system. Similarly to the $A/\varphi$ formulation [7], the weak continuous and discrete formulations are established, and the well-posedness of both of them is addressed. Some useful analytical tools are derived. Among them, an ad-hoc Helmholtz decomposition for the $T/\Omega$ case is derived, which allows to pertinently split the error. Consequently, an a posteriori error estimator is obtained, which is proven to be reliable and locally efficient. Finally, numerical tests confirm the theoretical results.

In this paper an implicit fully discrete local discontinuous Galerkin (LDG) finite element method is applied to solve the time-fractional seventh-order Korteweg-de Vries (sKdV) equation, which is introduced by replacing the integer-order time derivatives with fractional derivatives. We prove that our scheme is unconditional stable and $L^2$ error estimate for the linear case with the convergence rate $O(h^{k+1}+(\Delta t)^2+(\Delta t)^{\frac{\alpha}{2}}h^{k+\frac{1}{2}})$ through analysis. Extensive numerical results are provided to demonstrate the performance of the present method.

This article presents the convergence analysis of a sequence of piecewise constant and piecewise linear functions obtained by the Rothe method to the solution of the first order evolution partial differential inclusion $u'(t)+Au(t)+\iota^{*} ∂ J(\iota u(t)) \ni f(t)$, where the multivalued term is given by the Clarke subdifferential of a locally Lipschitz functional. The method provides the proof of existence of solutions alternative to the ones known in literature and together with any method for underlying elliptic problem, can serve as the effective tool to approximate the solution numerically. Presented approach puts into the unified framework known results for multivalued nonmonotone source term and boundary conditions, and generalizes them to the case where the multivalued term is defined on the arbitrary reflexive Banach space as long as appropriate conditions are satisfied. In addition, the results on improved convergence as well as the numerical examples are presented.

We present techniques of a posteriori error estimation for $Q_1$ finite element discretizations based on residual evaluations with respect to test functions of higher-order. This technique is designed for quadrilateral (or hexahedral) triangulations and gives local error indicators in terms of nodal contributions. We show reliability and efficiency of the estimator. Moreover, we present a simplification which is attractive from computational point of view as well.

In this paper, we study two fully discrete schemes for the equations of motion arising in the Kelvin-Voigt model of viscoelastic fluids. Based on a backward Euler method in time and a finite element method in spatial direction, optimal error estimates which exhibit the exponential decay property in time are derived. In the later part of this article, a second order two step backward difference scheme is applied for temporal discretization and again exponential decay in time for the discrete solution is discussed. Finally, a priori error estimates are derived and results on numerical experiments conforming theoretical results are established.