a divergence free reconstruction of the nonconforming virtual element method for the stokes problem abstract in this paper we propose and investigate a divergence free reconstruction of the nonconforming virtual element for the stokes problem by constructing the computable raviart thomas like interpolation operator we guarantee the independence between the velocity error estimation u u h 1 h and the continuous pressure p as it happens for the divergence free flow solver moreover this modified scheme can also inherit the advantages of the classical nonconforming virtual element method such as very general meshes including non convex and degenerate elements a unified scheme for an arbitrary order approximation accuracy k etc then we provide the optimal l 2 error estimates for the velocity gradient and the pressure by taking advantage of the raviart thomas like interpolation operator and avoiding the use of a trace inequality finally three numerical experiments are presented to conform the theoretical analysis

divergence conforming discontinuous galerkin finite elements for stokes eigenvalue problems in this paper we present a divergence conforming discontinuous galerkin finite element method for stokes eigenvalue problems we prove a priori error estimates for the eigenvalue and eigenfunction errors and present a residual based a posteriori error estimator the a posteriori error estimator is proven to be reliable and locally efficient we finally present some numerical examples that verify the a priori convergence rates and the reliability and efficiency of the residual based a posteriori error estimator

an embedded hybridized discontinuous galerkin finite element method for the stokes equations abstract we present and analyze a new embedded hybridized discontinuous galerkin finite element method for the stokes problem the method has the attractive properties of full hybridized methods namely an h div conforming velocity field pointwise satisfaction of the continuity equation and a priori error estimates for the velocity that are independent of the pressure the embedded hybridized formulation has advantages over a full hybridized formulation in that it has fewer global degrees of freedom for a given mesh and the algebraic structure of the resulting linear system is better suited to fast iterative solvers the analysis results are supported by a range of numerical examples that demonstrate rates of convergence and which show computational efficiency gains over a full hybridized formulation

high order algebraic splitting for magnetohydrodynamics simulation this paper proposes analyzes and tests high order algebraic splitting methods for magnetohydrodynamic mhd flows the main idea is to apply at each time step yosida type algebraic splitting to a block saddle point problem that arises from a particular incremental formulation of mhd by doing so we dramatically reduce the complexity of the nonsymmetric block schur complement by decoupling it into two stokes type schur complements each of which is symmetric positive definite and also is the same at each time step we prove the splitting is o t3 accurate and if used together with block pressure correction is fourth order a full analysis of the solver is given both as a linear algebraic approximation but also in a finite element context that uses the natural spatial norms numerical tests are given to illustrate the theory and show the effectiveness of the method

on really locking free mixed finite element methods for the transient incompressible stokes equations inf sup stable mixed methods for the steady incompressible stokes equations that relax the divergence constraint are often claimed to deliver locking free discretizations however this relaxation leads to a pressure dependent contribution in the velocity error which is proportional to the inverse of the viscosity thus giving rise to a different locking phenomenon however a recently proposed modification of the right hand side alone leads to a discretization that is really locking free i e its velocity error converges with optimal order and is independent of the pressure and the smallness of the viscosity in this contribution we extend this approach to the transient incompressible stokes equations where besides the right hand side also the velocity time derivative requires an improved space discretization semidiscrete and fully discrete a priori velocity and pressure error estimates are derived which show remarkable robustness properties two numerical examples illustrate the superior accura

exact hausdorff measure on the boundary of a galton watson tree a necessary and sufficient condition for the almost sure existence of an absolutely continuous with respect to the branching measure exact hausdorff measure on the boundary of a galton watson tree is obtained in the case where the absolutely continuous exact hausdorff measure does not exist almost surely a criterion which classifies gauge functions 0 according to whether 0 hausdorff measure of the boundary minus a certain exceptional set is zero or infinity is given important examples are discussed in four additional theorems in particular hawkes s conjecture in 1981 is solved problems of determining the exact local dimension of the branching measure at a typical point of the boundary are also solved

a universal chung type law of the iterated logarithm let x 1 x 2 be a sequence of independent and identically distributed random variables we find sequences of norming and centering constants n and n such that a universal chung type law of the iterated logarithm holds namely lim inf n max 1 k n s k k n n almost surely where s k denotes the sum of the first k of x 1 x 2 k 1 if the underlying distribution function is in the feller class we show that this lim inf is strictly positive with probability 1

on the determination of the grad div criterion abstract grad div stabilization adding a term grad div u to penalize violation of incompressibility has proven to be a useful tool in the simulation of incompressible flows such a term requires a choice of the coefficient and studies have begun appearing with various suggestions for its value we give an analysis herein that provides a restricted range of possible values for the coefficient in 3d turbulent flows away from walls if u l denote the large scale velocity and length respectively and is the signal to noise ratio of the body force estimates suggest that should be restricted to the range 2 24 l u 2 4 r e l u mesh independent case 2 24 l u 2 4 h l 4 3 l u mesh dependent case

a conservative finite element method for the incompressible euler equations with variable density abstract we construct a finite element discretization and time stepping scheme for the incompressible euler equations with variable density that exactly preserves total mass total squared density total energy and pointwise incompressibility the method uses raviart thomas or brezzi douglas marini finite elements to approximate the velocity and discontinuous polynomials to approximate the density and pressure to achieve exact preservation of the aforementioned conserved quantities we exploit a seldom used weak formulation of the momentum equation and a second order time stepping scheme that is similar but not identical to the midpoint rule we also describe and prove stability of an upwinded version of the method we present numerical examples that demonstrate the order of convergence of the method

a gradient robust well balanced scheme for the compressible isothermal stokes problem abstract a novel notion for constructing a well balanced scheme a gradient robust scheme is introduced and a showcase application for the steady compressible isothermal stokes equations in a nearly hydrostatic situation is presented gradient robustness means that gradient fields in the momentum balance are well balanced by the discrete pressure gradient which is possible on arbitrary unstructured grids the scheme is asymptotic preserving in the sense that it reduces for low mach numbers to a recent inf sup stable and pressure robust discretization for the incompressible stokes equations the convergence of the coupled fem fvm scheme for the nonlinear isothermal stokes equations is proved by compactness arguments numerical examples illustrate the numerical analysis and show that the novel approach can lead to a dramatically increased accuracy in nearly hydrostatic low mach number flows numerical examples also suggest that a straightforward extension to barotropic situations with nonlinear equations of state is feasible

on the divergence constraint in mixed finite element methods for incompressible flows the divergence constraint of the incompressible navier stokes equations is revisited in the mixed finite element framework while many stable and convergent mixed elements have been developed throughout the past four decades most classical methods relax the divergence constraint and only enforce the condition discretely as a result these methods introduce a pressure dependent consistency error which can potentially pollute the computed velocity these methods are not robust in the sense that a contribution from the right hand side which influences only the pressure in the continuous equations impacts both velocity and pressure in the discrete equations this article reviews the theory and practical implications of relaxing the divergence constraint several approaches for improving the discrete mass balance or even for computing divergence free solutions will be discussed grad div stabilization higher order mixed methods derived on the basis of an exact de rham complex h div conforming finite

a compatible high order meshless method for the stokes equations with applications to suspension flows abstract a stable numerical solution of the steady stokes problem requires compatibility between the choice of velocity and pressure approximation that has traditionally proven problematic for meshless methods in this work we present a discretization that couples a staggered scheme for pressure approximation with a divergence free velocity reconstruction to obtain an adaptive high order finite difference like discretization that can be efficiently solved with conventional algebraic multigrid techniques we use analytic benchmarks to demonstrate equal order convergence for both velocity and pressure when solving problems with curvilinear geometries in order to study problems in dense suspensions we couple the solution for the flow to the equations of motion for freely suspended particles in an implicit monolithic scheme the combination of high order accuracy with fully implicit schemes allows the accurate resolution of stiff lubrication forces directly from the solution of the stokes problem without the need to introduce sub grid lubrication models

pressure robustness and discrete helmholtz projectors in mixed finite element methods for the incompressible navier stokes equations abstract recently it was understood how to repair a certain l 2 orthogonality of discretely divergence free vector fields and gradient fields such that the velocity error of inf sup stable discretizations for the incompressible stokes equations becomes pressure independent these new pressure robust stokes discretizations deliver a small velocity error whenever the continuous velocity field can be well approximated on a given grid on the contrary classical inf sup stable stokes discretizations can guarantee a small velocity error only when both the velocity and the pressure field can be approximated well simultaneously in this contribution pressure robustness is extended to the time dependent navier stokes equations in particular steady and time dependent potential flows are shown to build an entire class of benchmarks where pressure robust discretizations can outperform classical approaches significantly speedups will be explained by a new theoretical concept the discrete helmholtz projector of an inf sup stable discretization moreover different discrete nonlinear convection terms are discussed and skew symmetric pressure robust discretizations are proposed

pole assignment problem in two dimensional linear systems sufficient conditions for existence of a solution to the pole assignment problem in a class of 2 d linear systems using state feedback are given an algorithm for finding the feedback gain matrix for a class of 2 d linear systems is presented and illustrated by simple numerical example

a unified framework of continuous and discontinuous galerkin methods for solving the incompressible navier stokes equation abstract in this paper we propose a unified numerical framework for the time dependent incompressible navier stokes equation which yields the h 1 h div conforming and discontinuous galerkin methods with the use of different viscous stress tensors and penalty terms for pressure robustness under minimum assumption on galerkin spaces the semi and fully discrete stability is proved when a family of implicit runge kutta methods are used for time discretization furthermore we present a unified discussion on the penalty term numerical experiments are presented to compare our schemes with classical schemes in the literature in both unsteady and steady situations it turns out that our scheme is competitive when applied to well known benchmark problems such as taylor green vortex kovasznay flow potential flow lid driven cavity flow and the flow around a cylinder

a discontinuous skeletal method for the viscosity dependent stokes problem 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 ge 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 l 2 pressure estimate of order k 1 and an l 2 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

virtual elements for the navier stokes problem on polygonal meshes a family of virtual element methods for the two dimensional navier stokes equations is proposed and analyzed the schemes provide a discrete velocity field which is pointwise divergence free a ri

the mini mixed finite element for the stokes problem an experimental investigation abstract o h 3 2 superconvergence in pressure and velocity has been experimentally investigated for the two dimensional stokes problem discretized with the mini mixed finite element even though the classic mixed finite element theory for the mini element guarantees linear convergence for the total error recent theoretical results indicate that superconvergence of order o h 3 2 in pressure and of the linear part of the computed velocity to the piecewise linear nodal interpolation of the exact velocity is in fact possible with structured three directional triangular meshes the numerical experiments presented here suggest a more general validity of o h 3 2 superconvergence possibly to automatically generated and unstructured triangulations in addition the approximating properties of the complete computed velocity have been compared with the approximating properties of the piecewise linear part of the computed velocity finding that the former is generally closer to the exact velocity whereas the latter conserves mass better

versatile mixed methods for the incompressible navier stokes equations abstract in the spirit of the principle of equipresence introduced by truesdell toupin the classical field theories 1960 we use the full version of the viscous stress tensor u u t 2 3 u i which was originally derived for compressible flows instead of the classical incompressible stress tensor u note that here is the dynamic viscosity coefficient and u is the velocity field in our approach the divergence free constraint for the viscous stress term is not enforced ahead of discretization instead our formulation allows the scheme itself to choose a consistent way to interpret the divergence free constraint i e the divergence free constraint is interpreted or enforced in a consistent fashion in both the mass conservation equation and the stress tensor term in the momentum equation furthermore our approach preserves the original symmetrical properties of the stress tensor e g its rotational invariance and it remains physically correct in the context of compressible flows as a result our approach facilitates versatility and code reuse in this paper we introduce our approach and establish some important mathematical properties for the resulting class of finite element schemes more precisely for general mixed methods which are not necessarily pointwise divergence free we establish the existence of a new norm induced by the full viscous bilinear form thereafter we prove the coercivity of the viscous bilinear form and the semi coercivity of a convective trilinear form in addition we demonstrate l2 stability of the discrete velocity fields for the general class of methods and by deduction the h div conforming methods finally we run some numerical experiments to illustrate the behavior of the versatile mixed methods and we make careful comparisons with a conventional h div conforming scheme

the analogue of grad div stabilization in dg methods for incompressible flows limiting behavior and extension to tensor product meshes abstract grad div stabilization is a classical remedy in conforming mixed finite element methods for incompressible flow problems for mitigating velocity errors that are sometimes called poor mass conservation such errors arise due to the relaxation of the divergence constraint in classical mixed methods and are excited whenever the spatial discretization has to deal with comparably large and complicated pressures in this contribution an analogue of grad div stabilization for discontinuous galerkin methods is studied here the key is the penalization of the jumps of the normal velocities over facets of the triangulation which controls the measure valued part of the distributional divergence of the discrete velocity solution our contribution is twofold first we characterize the limit for arbitrarily large penalization parameters which shows that the stabilized nonconforming discontinuous galerkin methods remain robust and accurate in this limit second we extend these ideas to the case of non simplicial meshes here broken grad div stabilization must be used in addition to the normal velocity jump penalization in order to get the desired pressure robustness effect the analysis is performed for the stokes equations and more complex flows and crouzeix raviart elements are considered in numerical examples that also show the relevance of the theory in practical settings

longer time accuracy for incompressible navier stokes simulations with the emac formulation abstract in this paper we consider the recently introduced emac formulation for the incompressible navier stokes ns equations which is the only known ns formulation that conserves energy momentum and angular momentum when the divergence constraint is only weakly enforced since its introduction the emac formulation has been successfully used for a wide variety of fluid dynamics problems we prove that discretizations using the emac formulation are potentially better than those built on the commonly used skew symmetric formulation by deriving a better longer time error estimate for emac while the classical results for schemes using the skew symmetric formulation have gronwall constants dependent on exp c r e t with r e the reynolds number it turns out that the emac error estimate is free from this explicit exponential dependence on the reynolds number additionally it is demonstrated how emac admits smaller lower bounds on its velocity error since incorrect treatment of linear momentum angular momentum and energy induces lower bounds for l 2 velocity error and emac treats these quantities more accurately results of numerical tests for channel flow past a cylinder and 2d kelvin helmholtz instability are also given both of which show that the advantages of emac over the skew symmetric formulation increase as the reynolds number gets larger and for longer simulation times

a c1 virtual element method on polyhedral meshes abstract the purpose of the present paper is to develop c 1 virtual elements in three dimensions for linear elliptic fourth order problems motivated by the difficulties that standard conforming finite elements encounter in this framework we focus the presentation on the lowest order case the generalization to higher orders being briefly provided in the appendix the degrees of freedom of the proposed scheme are only 4 per mesh vertex representing function values and gradient values interpolation error estimates for the proposed space are provided together with a set of numerical tests to validate the method at the practical level

analysis of algebraic chorin temam splitting for incompressible nse and comparison to yosida methods abstract algebraic splitting methods are a common approach to solving the saddle point linear systems that arise at each time step of an incompressible flow simulation there are two main classes of these methods those of yosida type and those of algebraic chorin temam act type with the yosida methods being predominantly used in practice we show herein through new analysis and extensive numerical testing that act methods that include the viscous term stiffness matrix in the modified a block are unconditionally stable and can be superior to yosida methods in a range of problems particular situations where the act type solvers are advantageous include problems where numerical stability is a concern as well as problems where strong enforcement of the divergence constraint is important

optimal spectral approximation of 2n order differential operators by mixed isogeometric analysis abstract we approximate the spectra of a class of 2 n order differential operators using isogeometric analysis in mixed formulations this class includes a wide range of differential operators such as those arising in elliptic biharmonic cahn hilliard swift hohenberg and phase field crystal equations the spectra of the differential operators are approximated by solving differential eigenvalue problems in mixed formulations which require auxiliary parameters the mixed isogeometric formulation when applying classical quadrature rules leads to an eigenvalue error convergence of order 2 p where p is the order of the underlying b spline space we improve this order to be 2 p 2 by applying optimally blended quadrature rules developed in puzyrev et al 2017 caloet al 0000 and this order is an optimum in the view of dispersion error we also compare these results with the mixed finite elements and show numerically that the mixed isogeometric analysis leads to significantly better spectral approximations

towards a unified finite element method for the stokes equations we present an algorithm for solving the stokes system based on the scott vogelius technique together with the iterated penalty method we show that by projecting the scott vogelius pressure onto

variational multiscale modeling with discretely divergence free subscales abstract we introduce a residual based stabilized formulation for incompressible navier stokes flow that maintains discrete and for divergence conforming methods strong mass conservation for inf sup stable spaces with h 1 conforming pressure approximation while providing optimal convergence in the diffusive regime robustness in the advective regime and energetic stability the method is formally derived using the variational multiscale vms concept but with a discrete fine scale pressure field which is solved for alongside the coarse scale unknowns such that the coarse and fine scale velocities separately satisfy discrete mass conservation we show energetic stability for the full navier stokes problem and we prove convergence and robustness for a linearized model oseen flow under the assumption of a divergence conforming discretization numerical results indicate that all properties extend to the fully nonlinear case and that the proposed formulation can serve to model unresolved turbulence

on reference solutions and the sensitivity of the 2d kelvin helmholtz instability problem abstract two dimensional kelvin helmholtz instability problems are popular examples for assessing discretizations for incompressible flows at high reynolds number unfortunately the results in the literature differ considerably this paper presents computational studies of a kelvin helmholtz instability problem with high order divergence free finite element methods reference results in several quantities of interest are obtained for three different reynolds numbers up to the beginning of the final vortex pairing a mesh independent prediction of the final pairing is not achieved due to the sensitivity of the considered problem with respect to small perturbations a theoretical explanation of this sensitivity to small perturbations is provided based on the theory of self organization of 2d turbulence possible sources of perturbations that arise in almost any numerical simulation are discussed

refined a posteriori error estimation for classical and pressure robust stokes finite element methods recent works showed that pressure robust modifications of mixed finite element methods for the stokes equations outperform their standard versions in many cases this is achieved by divergence free reconstruction operators and results in pressure independent velocity error estimates which are robust with respect to small viscosities in this paper we develop a posteriori error control which reflects this robustness the main difficulty lies in the volume contribution of the standard residual based approach that includes the l 2 norm of the right hand side however the velocity is only steered by the divergence free part of this source term an efficient error estimator must approximate this divergence free part in a proper manner otherwise it can be dominated by the pressure error to overcome this difficulty a novel approach is suggested that uses arguments from the stream function and vorticity formulation of the navier stokes equations the novel error estimators only take the hbox curl of the right hand side into account and so lead to provably reliable efficient and pressure independent upper bounds in case of a pressure robust method in particular in pressure dominant situations this is also confirmed by some numerical examples with the novel pressure robust modifications of the taylor hood and mini finite element methods

new mixed elements for maxwell equations new inf sup stable mixed elements are proposed and analyzed for solving the maxwell equations in terms of electric field and lagrange multiplier nodal continuous lagrange elements of any order on simplexes in two and three dimensional spaces can be used for the electric field the multiplier is compatibly approximated always by the discontinuous piecewise constant elements a general theory of stability and error estimates is developed when applied to the eigenvalue problem we show that the proposed mixed elements provide spectral correct spurious free approximations essentially optimal error bounds only up to an arbitrarily small constant are obtained for eigenvalues and for both singular and smooth solutions numerical experiments are performed to illustrate the theoretical results

an explicit divergence free dg method for incompressible flow abstract we present an explicit divergence free dg method for incompressible flow based on velocity formulation only an h div conforming and globally divergence free finite element space is used for the velocity field and the pressure field is eliminated from the equations by design the resulting ode system can be discretized using any explicit time stepping methods we use the third order strong stability preserving runge kutta method in our numerical experiments our spatial discretization produces the identical velocity field as the divergence conforming dg method of cockburn et al 2007 based on a velocity pressure formulation when the same dg operators are used for the convective and viscous parts due to the global nature of the divergence free constraint and its interplay with the boundary conditions it is very hard to construct local bases for our finite element space here we present a key result on the efficient implementation of the scheme by identifying the equivalence of the mass matrix inversion of the globally divergence free finite element space to a standard hybrid mixed poisson solver hence in each time step a hybrid mixed poisson solver is used which reflects the global nature of the incompressibility condition in the actual implementation of this fully discrete scheme the pressure field is also computed via the hybrid mixed poisson solver hence the scheme can be interpreted as a velocity pressure formulation that treats the incompressibility constraint and pressure forces implicitly but the viscous and convective part explicitly since we treat viscosity explicitly for the navier stokes equation our method shall be best suited for unsteady high reynolds number flows so that the cfl constraint is not too restrictive

global in time stability and accuracy of imex fem data assimilation schemes for navier stokes equations abstract we study numerical schemes for incompressible navier stokes equations using imex temporal discretizations finite element spatial discretizations and equipped with continuous data assimilation a technique recently developed by azouani et al 2014 we analyze stability and accuracy of the proposed methods and are able to prove well posedness long time stability and long time accuracy estimates under restrictions of the time step size and data assimilation parameter we give results for several numerical tests that illustrate the theory and show that for good results the choice of discretization parameter and element choices can be critical

a universal one sided law of the iterated logarithm we prove that the lim inf of suitably normalized sums of i i d nonnegative and nondegenerate random variables can with probability 1 only be a constant between 2 1 2 and 0 moreover we show that each value within this range is attainable by an appropriate choice of the underlying common distribution function