The use of catalytic membranes in chemical reactor engineering has significantly improved the reactor performance. Therefore, mathematical modelling and numerical simulations of catalytic membrane reactors are important for optimal parameter design increasing the selectivity and yield of certain products in chemical reactions. In this paper, the mathematical model of the flow-through catalytic membrane reactor is based on coupled nonlinear convection-diffusion-reaction equations with temperature-dependent reaction rate constants. The series reactions in the reactor are characterized by the power-law kinetics of fractional order which until now has been rarely investigated in the literature. Under specific reaction and process conditions the reactant can be depleted and a dead-zone can be formed in the membrane reactor. The numerical simulation of this phenomenon requires special solvers. The key idea of this work is to construct an appropriate time-marching scheme for solving the steady-state model equations. The effects of parameters such as reaction order, Peclet number, and Thiele modulus on solution profiles and formation of dead-zones are studied numerically. The simulation results show how these parameters influence the appearance and size of the dead zone for the non-isothermal multi-reaction systems.

We introduce and analyze a post-processing for a family of variational space-time approximations to wave problems. The discretization in space and time is based on continuous finite element methods. The post-processing lifts the fully discrete approximations in time from continuous to continuously differentiable ones. Further, it increases the order of convergence of the discretization in time which can be be exploited nicely, for instance, for a-posteriori error control. The convergence behavior is shown by proving error estimates of optimal order in various norms. A bound of superconvergence at the discrete times nodes is included. To show the error estimates, a special approach is developed. Firstly, error estimates for the time derivative of the post-processed solution are proved. Then, in a second step these results are used to establish the desired error estimates for the post-processed solution itself. The need for this approach comes through the structure of the wave equation providing only stability estimates that preclude us from using absorption arguments for the control of certain error quantities. A further key ingredient of this work is the construction of a new time-interpolate of the exact solution that is needed in an essential way for deriving the error estimates. Finally, a conservation of energy property is shown for the post-processed solution which is a key feature for approximation schemes to wave equations. The error estimates given in this work are confirmed by numerical experiments.

We construct L2-orthogonal conforming elements of arbitrary order for the Local Projection Stabilization (LPS). L2-orthogonal basis functions lead to a diagonal mass matrix which can be advantageous for time discretizations. We prove that the constructed family of finite elements satisfies a local inf-sup condition. Additionally, we investigate the size of the local inf-sup constant with respect to the polynomial degree. Our numerical tests show that the discrete solution is oscillation-free and of optimal accuracy in the regions away from the boundary or interior layers.

We analyze the discontinuous Galerkin method in time combined with a finite element method with symmetric stabilization in space to approximate evolution problems with a linear, first-order differential operator. A unified analysis is presented for space discretization, including the discontinuous Galerkin method and H-1-conforming finite elements with interior penalty on gradient jumps. Our main results are error estimates in various norms for smooth solutions. Two key ingredients are the post-processing of the fully discrete solution by lifting its jumps in time and a new time-interpolate of the exact solution. We first analyze the L-infinity(L-2) (at discrete time nodes) and L-2(L-2) errors and derive a superconvergent bound of order (tau(k+2) + h(r+1/2)) for static meshes for k >= 1. Here, tau is the time step, k the polynomial order in time, h the size of the space mesh, and r the polynomial order in space. For the case of dynamically changing meshes, we derive a novel bound on the resulting projection error. Finally, we prove new optimal bounds on static meshes for the error in the time-derivative and in the discrete graph norm.

We propose a continuous interior penalty (CIP) method for the pure transport problem and for the viscosity dependent "Stokes-Brinkman" problem where the gradient jump penalty is localized to faces in the interior of subdomains. Special focus is given to the case where the subdomains are so-called composite finite elements, e.g., quadrilateral, hexahedral or prismatic elements which are composed by simplices such that the arising global simplicial mesh is regular. The advantage of this local CIP is that it allows for static condensation in contrast to the classical CIP method. If the degrees of freedom in the interior of the composite finite elements are eliminated using static condensation then the resulting couplings of the skeleton degrees of freedom are comparable to those for classical conforming finite element methods which leads to a substantially smaller matrix stencil than for the standard global CIP method. Optimal stability and error estimates are proved and numerical tests are presented. For the Stokes-Brinkman model, our error bound does not increase if the viscosity parameter tends to zero which is mainly achieved by adding a penalty term for the divergence of the velocity in the discretization. Moreover, the reduction effect of the static condensation is much stronger for this model since, beside the elimination of all velocity degrees of freedom in the interior of each composite cell, all pressure degrees of freedom except for the cellwise constants can be eliminated.

We devise and analyze arbitrary-order nonconforming methods for the discretization of the viscosity-dependent Stokes equations on simplicial meshes. We keep track explicitly of the viscosity and aim at pressure-robust schemes that can deal with the practically relevant case of body forces with large curl-free part in a way that the discrete velocity error is not spoiled by large pressures. The method is inspired from the recent Hybrid High-Order (HHO) methods for linear elasticity. After elimination of the auxiliary variables by static condensation, the linear system to be solved involves only discrete face-based velocities, which are polynomials of degree k≥0, and cell-wise constant pressures. Our main result is a pressure-independent energy-error estimate on the velocity of order (k+1). The main ingredient to achieve pressure-independence is the use of a divergence-preserving velocity reconstruction operator in the discretization of the body forces. We also prove an L2-pressure estimate of order (k+1) and an L2-velocity estimate of order (k+2), the latter under elliptic regularity. The local mass and momentum conservation properties of the discretization are also established. Finally, two- and three-dimensional numerical results are presented to support the analysis.

We consider the Willmore flow of two-dimensional graphs subject to Dirichlet boundary conditions. The corresponding evolution is described by a highly nonlinear parabolic PDE of fourth order for the height function. Based on a suitable weak form of the equation we derive a semidiscrete scheme which uses C-1-finite elements and interpolates the Dirichlet boundary conditions. We prove quasioptimal error bounds in Sobolev norms for the solution and its time derivative and present results of test calculations.

The local projection stabilization (LPS) method is an attractive space discretization technique for non-stationary convection-dominated transport problems. Combined with an implicit time discretization, it yields a stable and non-oscillatory high-order finite element approximation in regions where the exact solution is sufficiently smooth. However, the accuracy of the LPS solution deteriorates in the neighborhood of discontinuities or steep gradients. Like any other linear high-order scheme, LPS tends to produce spurious oscillations in these regions. The usual way to avoid such local oscillations is to add some shock-capturing terms acting as nonlinear artificial viscosity. In many cases, this leads to marked improvements but such good results depend on the right choice of a free parameter. In contrast to traditional shock capturing, the flux-corrected transport (FCT) methodology makes it possible to prevent numerical oscillations and to enforce the discrete maximum principle. FCT can be regarded as a technique for blending a stable high-order discretization with a monotone low-order discretization which contains enough artificial diffusion to suppress undershoots and overshoots. To this end, the difference between the two approximations is decomposed into antidiffusive fluxes or element contributions which are multiplied by solutiondependent correction factors. The FCT solution is guaranteed to be non-oscillatory everywhere and to revert to the underlying high-order approximation in smooth regions. In the finite element literature, the FCT method has successfully been used to combine an unstable high-order Galerkin scheme with a low-order upwind-biased discretization. Since the unconstrained Galerkin approximation may exhibit global oscillations, the FCT limiter may need to be activated everywhere, which may destroy the high accuracy in regions of smoothness. This concern has led us to combine the stable high-order LPS discretization with a loworder artificial viscosity method. Since the LPS method is linearly stable, the FCT correction of antidiffusive fluxes is restricted to small subdomains, whereas optimal accuracy is maintained elsewhere. In this talk, we discuss the practical implementation of the proposed FCT method in the case of one space dimension and present some numerical results.

In this contribution, we show a method for the boundary feedback stabilization of the Stokes problem around a stationary trajectory. We derive a formal low-rank algorithm for solving the stabilization problem in operator notation. The appearing operator equations are formulated in terms of stationary partial differential equations (PDEs) instead of using their finite dimensional representations in terms of matrices. A Galerkin method, satisfying the divergence constraint pointwise locally is especially appealing since it represents appropriately the action of the Helmholtz projection.The main advantages of the composite technique are the efficient assembly of element matrices, the reduction of computational costs using static condensation, and the diagonal mass matrix. The non-conforming character of the composite element guarantees a better sparsity pattern, compared to conforming elements, due to the lower number of couplings between basis functions corresponding to neighboring cells. We also achieve the pointwise mass conservation on sub-triangles of each element.

In this paper, we discuss solution techniques of Newton-multigrid type for the resulting nonlinear saddle-point block-systems if higher order continuous Galerkin–Petrov (cGP(k)) and discontinuous Galerkin (dG(k)) time discretizations are applied to the nonstationary incompressible Navier–Stokes equations. In particular for the cGP(2) method with quadratic ansatz functions in time, which lead to 3rd order accuracy in the L2-norm and even to 4th order superconvergence in the endpoints of the time intervals, together with the finite element pair Q2/P1disc for the spatial approximation of velocity and pressure leading to a globally 3rd order scheme, we explain the algorithmic details as well as implementation aspects. All presented solvers are analyzed with respect to their numerical costs for two prototypical flow configurations.

This paper presents a postprocessing technique for estimating the local regularity of numerical solutions in high-resolution finite element schemes. A derivative of degree p ≥ 0 is considered to be smooth if a discontinuous linear reconstruction does not create new maxima or minima. The intended use of this criterion is the identification of smooth cells in the context of p-adaptation or selective flux limiting. As a model problem, we consider a 2D convection equation discretized with bilinear finite elements. The discrete maximum principle is enforced using a linearized flux-corrected transport algorithm. The deactivation of the flux limiter in regions of high regularity makes it possible to avoid the peak clipping effect at smooth extrema without generating spurious undershoots or overshoots elsewhere.

SUMMARYIn this paper, we present fully implicit continuous Galerkin–Petrov (cGP) and discontinuous Galerkin (dG) time‐stepping schemes for incompressible flow problems which are, in contrast to standard approaches like for instance the Crank–Nicolson scheme, of higher order in time. In particular, we analyze numerically the higher order dG(1) and cGP(2) methods, which are super convergent of third, resp., fourth order in time, whereas for the space discretization, the well‐known LBB‐stable finite element pair of third‐order accuracy is used. The discretized systems of nonlinear equations are treated by using the Newton method, and the associated linear subproblems are solved by means of a monolithic (geometrical) multigrid method with a blockwise Vanka‐like smoother treating all components simultaneously. We perform nonstationary simulations (in 2D) for two benchmarking configurations to analyze the temporal accuracy and efficiency of the presented time discretization schemes w.r.t. CPU and numerical costs. As a first test problem, we consider a classical ‘flow around cylinder’ benchmark. Here, we concentrate on the nonstationary behavior of the flow patterns with periodic oscillations and examine the ability of the different time discretization schemes to capture the dynamics of the flow. As a second test case, we consider the nonstationary ‘flow through a Venturi pipe’. The objective of this simulation is to control the instantaneous and mean flux through this device. Copyright © 2013 John Wiley & Sons, Ltd.

In this paper, we extend our work for the heat equation in [4] a nd for the Stokes equations in [5] to the nonstationary Navier-Stokes equations in two dimensions. We examinecontinuousGalerkin-Petrov (cGP) time discretization schemes for nonstationary incompressible flow. In particular, we im plement and analyze numerically the higher order cGP(2)-method. For the space dis cretization, we use the LBB-stable finite element pair Q2/P 1 . The discretized systems of nonlinear equations are treated by using the fixed-point as well as the Newto n method and the associated linear subproblems are solved by using a monolithi c multigrid solver with GMRES method as smoother. We perform nonstationary simulat ions for a benchmarking configuration to analyze the temporal accuracy and e fficiency of the presented time discretization scheme.

AbstractWe consider a time-dependent convection diffusion equation in the transport dominated case. As a stabilization method in space we propose a new variant of Local Projection Stabilization (LPS) which uses special enriched bubble functions such that L²-orthogonal local basis functions can be constructed. L²-orthogonal basis functions lead to a diagonal mass matrix which is advantageous for time discretization. We use the discontinuous Galerkin method of polynomial order one for the discretization in time which is superconvergent of order three at the endpoints of the time intervals. In order to avoid the remaining oscillations in the LPS-solution we add for each time step in the space discretization an extra shock capturing term which acts only locally on those mesh cells where an error-indicator is relatively large. The novelty in the shock capturing term is that the scaling factor in front of the additive diffusion term is computed from a low order post-processing error. As a result we obtain both, an oscillation-free discrete solution and the information about the local regions where this solution is still inaccurate due to some smearing. The latter information can be used to create in each time step an adaptively refined space mesh. Whereas the numerical experiments are restricted to one space dimension the proposed ideas work also in the multi-dimensional spatial case. The numerical tests show that the discrete solution with shock capturing is oscillation-free and of optimal accuracy in the regions outside of the shock.

We show that existing quadrilateral nonconforming finite elements of higher order exhibit a reduction in the order of approximation if the sequence of meshes is still shape-regular but consists no longer of asymptotically affine equivalent mesh cells. We study second order nonconforming finite elements as members of a new family of higher order approaches which prevent this order reduction. We present a new approach based on the enrichment of the original polynomial space on the reference element by means of nonconforming cell bubble functions which can be removed at the end by static condensation. Optimal estimates of the approximation and consistency error are shown in the case of a Poisson problem which imply an optimal order of the discretization error. Moreover, we discuss the known nonparametric approach to prevent the order reduction in the case of higher order elements, where the basis functions are defined as polynomials on the original mesh cell. Regarding the efficient treatment of the resulting linear discrete systems, we analyze numerically the convergence of the corresponding geometrical multigrid solvers which are based on the canonical full order grid transfer operators. Based on several benchmark configurations, for scalar Poisson problems as well as for the incompressible Navier–Stokes equations (representing the desired application field of these nonconforming finite elements), we demonstrate the high numerical accuracy, flexibility and efficiency of the discussed new approaches which have been successfully implemented in the FeatFlow software (www.featflow.de). The presented results show that the proposed FEM-multigrid combinations (together with discontinuous pressure approximations) appear to be very advantageous candidates for efficient simulation tools, particularly for incompressible flow problems.

We consider composite non-conforming elements which are suitable for solving both, convection-diffusion and Oseen equations, respectively. Numerical experiments show that the usual Local Projection Stabilization (LPS) fails in the case of non-conforming elements. This drawback can be cured by adding a modified LPS-term to the discrete bilinear form. In contrast to the known one-level LPS approach, the proposed discrete space of composite non-conforming elements does not need special enrichments. One advantage of the composite non-conforming elements compared to known conforming quadrilateral elements like the Taylor-Hood elements is that they guarantee a better local mass conservation. Another advantage is that the matrices for the coupling between pressure and velocity are much more sparse for the composite non-conforming elements. Furthermore, the low order non-conforming element (Crouzeix-Raviart element) leads to a diagonal mass matrix which is advantageous for time dependent problems. We present several numerical experiments which show optimal approximation and good stability properties for our proposed non-conforming elements and the modified LPS.

In this paper, we extend our work for the heat equation in (Hussain et al., J Numer Math 19(1):41–61, 2011) and for the Stokes equations in (Hussain et al., Open Numer Methods J 4:35–45, 2012) to the nonstationary Navier-Stokes equations in two dimensions. We examine continuous Galerkin-Petrov (cGP) time discretization schemes for nonstationary incompressible flow. In particular, we implement and analyze numerically the higher order cGP(2)-method. For the space discretization, we use the LBB-stable finite element pair \(Q_{2}/P_{1}^{\mathit{disc}}\). The discretized systems of nonlinear equations are treated by using the fixed-point as well as the Newton method and the associated linear subproblems are solved by using a monolithic multigrid solver with GMRES method as smoother. We perform nonstationary simulations for a benchmarking configuration to analyze the temporal accuracy and efficiency of the presented time discretization scheme.

For the Darcy–Brinkman equations, which model porous media flow, we present an equal-order H1-conforming finite element method for approximating velocity and pressure based on a local projection stabilization technique. The method is stable and accurate uniformly with respect to the coefficients of the viscosity and the zeroth order term in the momentum equation. We prove a priori error estimates in a mesh-dependent norm as well as in the L2-norm for velocity and pressure. In particular, we obtain optimal order of convergence in L2 for the pressure in the Darcy case with vanishing viscosity and for the velocity in the general case with a positive viscosity coefficient. Numerical results for different values of the coefficients in the Darcy–Brinkman model are presented which confirm the theoretical results and indicate nearly optimal order also in cases which are not covered by the theory.

Starting from the known continuous Galerkin-Petrov (cGP) and discontinuous Galerkin (dG) time discretization method of some polynomial order k, we propose two new variational time discretizations for the system of ordinary differential equations (ODE) associated with the semi-discrete finite element solution of a parabolic partial differential equation. Both methods are based on a time polynomial ansatz of higher order k + 1 where the global smoothness of the discrete solution is also one level higher than that of the original method cGP or dG, respectively. Therefore, the total number of unknowns and the computational costs are not increasing but the accuracy is improved by one order. For the new methods, we prove optimal order a priori error estimates in the maximum norm for a general nonlinear ODE-system with a Lipschitz-continuous right-hand side. We show that the C-continuous Galerkin-Petrov methods are A-stable and that the Ccontinuous dG-methods are strongly A-stable (L-stable). Moreover, we prove that, in each case, the new higher order method and the original method coincide at the endpoints of the time intervals and that their difference can be computed by a simple post-processing step with low computational costs. With this relationship we have proven in the nonlinear case that, in particular, the cGP(2)and dG(1)-method are superconvergent of order 4 and 3, respectively, at the endpoints of the time intervals. Finally, we present numerical results for the Burgers equation which confirm the theoretical results.

We consider the Willmore boundary value problem for surfaces of revolution where, as Dirichlet boundary conditions, any symmetric set of position and angle may be prescribed. Using direct methods of the calculus of variations, we prove existence and regularity of minimising solutions. Moreover, we estimate the optimal Willmore energy and prove a number of qualitative properties of these solutions. Besides convexity-related properties we study in particular the limit when the radii of the boundary circles converge to 0, while the "length" of the surfaces of revolution is kept fixed. This singular limit is shown to be the sphere, irrespective of the prescribed boundary angles.These analytical investigations are complemented by presenting a numerical algorithm, based on C-1-elements, and numerical studies. They intensively interact with geometric constructions in finding suitable minimising sequences for the Willmore functional.