Many problems with underlying variational structure involve a coupling of volume with surface effects. A straight-forward approach in a finite element discretization is to make use of the surface triangulation that is naturally induced by the volume triangulation. In an adaptive method one wants to facilitate "matching" local mesh modifications, i.e., local refinement and/or coarsening, of volume and surface mesh with standard tools such that the surface grid is always induced by the volume grid. We describe the concepts behind this approach for bisectional refinement and describe new tools incorporated in the finite element toolbox ALBERTA. We also present several important applications of the mesh coupling.

In this paper, we present a $\mathbb{P}_N × \mathbb{P}_N$ spectral element method and a detailed comparison with existing methods for the unsteady incompressible Navier-Stokes equations. The main purpose of this work consists of: (i) detailed comparison and discussion of some recent developments of the temporal discretizations in the frame of spectral element approaches in space; (ii) construction of a stable $\mathbb{P}_N × \mathbb{P}_N$ method together with a $\mathbb{P}_N → \mathbb{P}_{N-2}$ post-filtering. The link of different methods will be clarified. The key feature of our method lies in that only one grid is needed for both velocity and pressure variables, which differs from most well-known solvers for the Navier-Stokes equations. Although not yet proven by rigorous theoretical analysis, the stability and accuracy of this one-grid spectral method are demonstrated by a series of numerical experiments.

In this paper, we investigate a streamline diffusion finite element approximation scheme for the constrained optimal control problem governed by linear convection dominated diffusion equations. We prove the existence and uniqueness of the discretized scheme. Then a priori and a posteriori error estimates are derived for the state, the co-state and the control. Three numerical examples are presented to illustrate our theoretical results.

This study was suggested by previous work on the simulation of evolution equations with scale-dependent processes, e.g., wave-propagation or heat-transfer, that are modeled by wave equations or heat equations. Here, we study both parabolic and hyperbolic equations. We focus on ADI (alternating direction implicit) methods and LOD (locally one-dimensional) methods, which are standard splitting methods of lower order, e.g. second-order. Our aim is to develop higher-order ADI methods, which are performed by Richardson extrapolation, Crank-Nicolson methods and higher-order LOD methods, based on locally higher-order methods. We discuss the new theoretical results of the stability and consistency of the ADI methods. The main idea is to apply a higher-order time discretization and combine it with the ADI methods. We also discuss the discretization and splitting methods for first-order and second-order evolution equations. The stability analysis is given for the ADI method for first-order time derivatives and for the LOD (locally one-dimensional) methods for second-order time derivatives. The higher-order methods are unconditionally stable. Some numerical experiments verify our results.

Let $L^2([0,1],x)$ be the space of the real valued, measurable, square summable functions on $[0,1]$ with weight $x$, and let $\mathcal{L}_n$ be the subspace of $L^2([0,1],x)$ defined by a linear combination of $J_0(\mu_kx)$, where $ J_0 $ is the Bessel function of order $0$ and $\{\mu_k\}$ is the strictly increasing sequence of all positive zeros of $J_0$. For $f\in L^2([0,1],x),$ let $E(f,{\cal L}_n)$ be the error of the best $L^2([0,1],x)$, i.e., approximation of $f$ by elements of ${\cal L}_n.$ The shift operator of $f$ at point $x\in [0, 1]$ with step $t\in [0, 1]$ is defined by$$T(t)f(x)=\frac{1}{\pi}\int_0^{\pi}f\big(\sqrt{x^2+t^2-2xt\cosθ}\big)dθ.$$The differences $(I-T(t))^{r/2}f=\sum_{j=0}^{\infty}(-1)^j{{r/2}\choose{j}}T^j(t)f$ of order $r\in (0, \infty)$ and the $L^2([0,1],x)-$modulus of continuity $\omega_r(f,\tau)=\sup\{\|(I-T(t))^{r/2}f\|:$ $0 \leq t\leq \tau \}$ of order $r$ are defined in the standard way, where $T^0(t)=I$ is the identity operator. In this paper, we establish the sharp Jackson inequality between $E(f, {\cal L}_n)$ and $\omega_r(f,\tau)$ for some cases of $r$ and $\tau$. More precisely, we will find the smallest constant $\mathcal{K}_n(\tau, r)$ which depends only on $n, r, $and $\tau,$ such that the inequality $E(f, {\cal L}_n)\le\mathcal{K}_n(\tau, r)(\tau, r)\omega_r(f,\tau)$ is valid.