A numerical scheme for the Reissner–Mindlin plate model is proposed. The method is based on a discrete Helmholtz decomposition and can be viewed as a generalization of the nonconforming finite element scheme of Arnold and Falk [SIAM J. Numer. Anal., 26(6):1276-1290, 1989]. The two unknowns in the discrete formulation are the in-plane rotations and the gradient of the vertical displacement. The decomposition of the discrete shear variable leads to equivalence with the usual Stokes system with penalty term plus two Poisson equations and the proposed method is equivalent to a stabilized discretization of the Stokes system that generalizes the Mini element. The method is proved to satisfy a best-approximation result which is robust with respect to the thickness parameter $t$.

We consider a control-constrained parabolic optimal control problem without Tikhonov term in the tracking functional. For the numerical treatment, we use variational discretization of its Tikhonov regularization: For the state and the adjoint equation, we apply Petrov-Galerkin schemes in time and usual conforming finite elements in space. We prove a-priori estimates for the error between the discretized regularized problem and the limit problem. Since these estimates are not robust if the regularization parameter tends to zero, we establish robust estimates, which — depending on the problem's regularity — enhance the previous ones. In the special case of bang-bang solutions, these estimates are further improved. A numerical example confirms our analytical findings.

We consider a strongly heterogeneous medium saturated by an incompressible viscous fluid as it appears in geomechanical modeling. This poroelasticity problem suffers from rapidly oscillating material parameters, which calls for a thorough numerical treatment. In this paper, we propose a method based on the local orthogonal decomposition technique and motivated by a similar approach used for linear thermoelasticity. Therein, local corrector problems are constructed in line with the static equations, whereas we propose to consider the full system. This allows benefiting from the given saddle point structure and results in two decoupled corrector problems for the displacement and the pressure. We prove the optimal first-order convergence of this method and verify the result by numerical experiments.

This paper proposes an efficient ADER (Arbitrary DERivatives in space and time) discontinuous Galerkin (DG) scheme to directly solve the Hamilton-Jacobi equation. Unlike multi-stage Runge-Kutta methods used in the Runge-Kutta DG (RKDG) schemes, the ADER scheme is one-stage in time discretization, which is desirable in many applications. The ADER scheme used here relies on a local continuous spacetime Galerkin predictor instead of the usual Cauchy-Kovalewski procedure to achieve high order accuracy both in space and time. In such predictor step, a local Cauchy problem in each cell is solved based on a weak formulation of the original equations in spacetime. The resulting spacetime representation of the numerical solution provides the temporal accuracy that matches the spatial accuracy of the underlying DG solution. The scheme is formulated in the modal space and the volume integral and the numerical fluxes at the cell interfaces can be explicitly written. The explicit formulae of the scheme at third order is provided on two-dimensional structured meshes. The computational complexity of the ADER-DG scheme is compared to that of the RKDG scheme. Numerical experiments are also provided to demonstrate the accuracy and efficiency of our scheme.

We design and numerically validate a recovery based linear finite element method for solving the biharmonic equation. The main idea is to replace the gradient operator $\nabla$ on linear finite element space by $G(\nabla)$ in the weak formulation of the biharmonic equation, where $G$ is the recovery operator which recovers the piecewise constant function into the linear finite element space. By operator $G$, Laplace operator $\Delta$ is replaced by $\nabla\cdot G(\nabla)$. Furthermore, the boundary condition on normal derivative $\nabla u\cdot \pmb{n}$ is treated by the boundary penalty method. The explicit matrix expression of the proposed method is also introduced. Numerical examples on the uniform and adaptive meshes are presented to illustrate the correctness and effectiveness of the proposed method.

We propose a class of 12 degrees of freedom triangular plate bending elements with quadratic rate of convergence. They may be viewed as the second order Specht triangle, while the Specht triangle is one of the best first order plate bending elements. The convergence result is proved under minimal smoothness assumption on the solution. Numerical results for both the smooth solution and nonsmooth solution confirm the theoretical prediction.

Pressure correction methods constitute the most widely used solvers for the time-dependent Navier-Stokes equations. There are several different pressure correction methods, where each time step usually consists in a predictor step for a non-divergence-free velocity, followed by a Poisson problem for the pressure (or pressure update), and a final velocity correction to obtain a divergence-free vector field. In some situations, the equations for the velocities are solved explicitly, so that the most numerically expensive step is the elliptic pressure problem. We here propose to solve this Poisson problem by a domain decomposition method which does not need any communication between the sub-regions. Hence, this system is perfectly adapted for parallel computation. We show under certain assumptions that this new scheme has the same order of convergence as the original pressure correction scheme (with  global projection). Numerical examples for the Stokes system show the effectivity of this new pressure correction method. The convergence order $\mathcal O(k^{ 2})$ for resulting velocity fields can be observed in the norm $l^2(0,T ;L^2(\Omega))$.

Optimal convergence rates of adaptive finite element methods are well understood in terms of the axioms of adaptivity. One key ingredient is the discrete reliability of a residual-based a posteriori error estimator, which controls the error of two discrete finite element solutions based on two nested triangulations. In the error analysis of nonconforming finite element methods, like the Crouzeix-Raviart or Morley finite element schemes, the difference of the piecewise derivatives of discontinuous approximations to the distributional gradients of global Sobolev functions plays a dominant role and is the object of this paper. The nonconforming interpolation operator, which comes naturally with the definition of the aforementioned nonconforming finite element in the sense of Ciarlet, allows for stability and approximation properties that enable direct proofs of the reliability for the residual that monitors the equilibrium condition. The novel approach of this paper is the suggestion of a right-inverse of this interpolation operator in conforming piecewise polynomials to design a nonconforming approximation of a given coarse-grid approximation on a refined triangulation. The results of this paper allow for simple proofs of the discrete reliability in any space dimension and multiply connected domains on general shape-regular triangulations beyond newest-vertex bisection of simplices. Particular attention is on optimal constants in some standard discrete estimates listed in the appendices.

The goal of this paper is to present a numerical method for the Smoluchowski equation, a drift-diffusion equation on the sphere, arising in the modelling of particle dynamics. The numerical method uses radial basis functions (RBF). This is a relatively new approach, which has recently mainly been used for geophysical applications. For a simplified model problem we compare the RBF approach with a spectral method, i.e. the standard approach used in related physical applications. This comparison as well as our other accuracy studies shows that RBF methods are an attractive alternative for this kind of models.

Under two hypotheses of nonconforming finite elements of fourth order elliptic problems, we present a side–patchwise projection based error analysis method (SPP–BEAM for short). Such a method is able to avoid both the regularity condition of exact solutions in the classical error analysis method and the complicated bubble function technique in the recent medius error analysis method. In addition, it is universal enough to admit generalizations. Then, we propose a sufficient condition for these hypotheses by imposing a set of in some sense necessary degrees of freedom of the shape function spaces. As an application, we use the theory to design a $P_3$ second order triangular $H^2$ non-conforming element by enriching two $P_4$ bubble functions and, another $P_4$ second order triangular $H^2$ nonconforming finite element, and a $P_3$ second order tetrahedral $H^2$ non-conforming element by enriching eight $P_4$ bubble functions, adding some more degrees of freedom.

We analyze the superconvergence property of the linear finite element method based on the polynomial preserving recovery (PPR) for Robin boundary elliptic problems on triangulations. First, we improve the convergence rate between the finite element solution and the linear interpolation under the $H^1$-norm by introducing a class of meshes satisfying the $Condition$ $(\alpha,\sigma,\mu)$. Then we prove the superconvergence of the recovered gradients post-processed by PPR and define an asymptotically exact a posteriori error estimator. Finally, numerical tests are provided to verify the theoretical findings.