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Continuous interior penalty finite element method for Oseen’s equations
 SIAM J. Numer. Anal
"... Abstract. In this paper we present an extension of the continuous interior penalty method of Douglas and Dupont [Interior penalty procedures for elliptic and parabolic Galerkin methods, in Computing Methods in Applied Sciences, Lecture Notes in Phys. 58, SpringerVerlag, Berlin, 1976, pp. 207–216] t ..."
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Cited by 36 (8 self)
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Abstract. In this paper we present an extension of the continuous interior penalty method of Douglas and Dupont [Interior penalty procedures for elliptic and parabolic Galerkin methods, in Computing Methods in Applied Sciences, Lecture Notes in Phys. 58, SpringerVerlag, Berlin, 1976, pp. 207–216] to Oseen’s equations. The method consists of a stabilized Galerkin formulation using equal order interpolation for pressure and velocity. To counter instabilities due to the pressure/velocity coupling, or due to a high local Reynolds number, we add a stabilization term giving L2control of the jump of the gradient over element faces (edges in two dimensions) to the standard Galerkin formulation. Boundary conditions are imposed in a weak sense using a consistent penalty formulation due to Nitsche. We prove energytype a priori error estimates independent of the local Reynolds number and give some numerical examples recovering the theoretical results.
CONTINUOUS INTERIOR PENALTY hpFINITE ELEMENT METHODS FOR ADVECTION AND ADVECTIONDIFFUSION EQUATIONS
"... Abstract. A continuous interior penalty hpfinite element method that penalizes the jump of the gradient of the discrete solution across mesh interfaces is introduced. Error estimates are obtained for advection and advectiondiffusion equations. The analysis relies on three technical results that ar ..."
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Cited by 29 (10 self)
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Abstract. A continuous interior penalty hpfinite element method that penalizes the jump of the gradient of the discrete solution across mesh interfaces is introduced. Error estimates are obtained for advection and advectiondiffusion equations. The analysis relies on three technical results that are of independent interest: an hpinverse trace inequality, a local discontinuous to continuous hpinterpolation result, and hperrorestimatesforcontinuousL 2orthogonal projections. 1.
On spurious oscillations at layers diminishing (SOLD) methods for convection–diffusion equations:
 Part I – A review. Comput. Methods Appl. Mech. Engrg.
, 2007
"... Abstract An unwelcome feature of the popular streamline upwind/PetrovGalerkin (SUPG) stabilization of convectiondominated convectiondiffusion equations is the presence of spurious oscillations at layers. Since the mid of the 1980s, a number of methods have been proposed to remove or, at least, t ..."
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Cited by 26 (11 self)
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Abstract An unwelcome feature of the popular streamline upwind/PetrovGalerkin (SUPG) stabilization of convectiondominated convectiondiffusion equations is the presence of spurious oscillations at layers. Since the mid of the 1980s, a number of methods have been proposed to remove or, at least, to diminish these oscillations without leading to excessive smearing of the layers. The paper gives a review and state of the art of these methods, discusses their derivation, proposes some alternative choices of parameters in the methods and categorizes them. Some numerical studies which supplement this review provide a first insight into the advantages and drawbacks of the methods.
A Stabilized Nonconforming Finite Element Method For Incompressible Flow
, 2004
"... In this paper we extend the recently introduced edge stabilization method to the case of nonconforming finite element approximations of the linearized NavierStokes equation. To get stability also in the convective dominated regime we add a term giving control of the jump in the gradient over el ..."
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Cited by 11 (3 self)
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In this paper we extend the recently introduced edge stabilization method to the case of nonconforming finite element approximations of the linearized NavierStokes equation. To get stability also in the convective dominated regime we add a term giving control of the jump in the gradient over element boundaries. An a priori error estimate that is uniform in the Reynolds number is proved and some numerical examples are presented. 1.
Preasymptotic error analysis of CIPFEM and FEM for the Helmholtz equation with high wave number. Part I: linear version
, 2014
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Weighted error estimates of the continuous interior penalty method for singularly perturbed problems
 IMA J. of Num. Anal
"... Abstract. In this paper we analyze local properties of the Continuous Interior Penalty (CIP) Method for a model convectiondominated singularly perturbed convectiondiffusion problem. We show weighted a priori error estimates, where the weight function exponentially decays outside the subdomain o ..."
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Cited by 7 (2 self)
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Abstract. In this paper we analyze local properties of the Continuous Interior Penalty (CIP) Method for a model convectiondominated singularly perturbed convectiondiffusion problem. We show weighted a priori error estimates, where the weight function exponentially decays outside the subdomain of interest. This result shows that locally, the CIP method is comparable to the Streamline Diffusion (SD) or the Discontinuous Galerkin (DG) methods. 1.
Explicit RungeKutta schemes and finite elements with symmetric stabilization for firstorder linear PDE systems
 SIAM J. NUMER. ANAL
, 2009
"... We analyze explicit Runge–Kutta schemes in time combined with stabilized finite elements in space to approximate evolution problems with a firstorder linear differential operator in space of Friedrichstype. For the time discretization, we consider explicit second and thirdorder Runge–Kutta sch ..."
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Cited by 7 (2 self)
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We analyze explicit Runge–Kutta schemes in time combined with stabilized finite elements in space to approximate evolution problems with a firstorder linear differential operator in space of Friedrichstype. For the time discretization, we consider explicit second and thirdorder Runge–Kutta schemes. We identify a general set of properties on the spatial stabilization, encompassing continuous and discontinuous finite elements, under which we prove stability estimates using energy arguments. Then, we establish L²norm error estimates with (quasi)optimal convergence rates for smooth solutions in space and time. These results hold under the usual CFL condition for thirdorder Runge–Kutta schemes and any polynomial degree in space and for secondorder Runge– Kutta schemes and firstorder polynomials in space. For secondorder Runge–Kutta schemes and higher polynomial degrees in space, a tightened 4/3CFL condition is required. Numerical results are presented for the advection and wave equations.
On the role of boundary conditions for CIP stabilization of higher order finite elements
 Electron. Trans. Numer. Anal
"... Abstract. We investigate the Continuous Interior Penalty (CIP) stabilization method for higher order nite elements applied to a convection diffusion equation with a small diffusion parameter . Performing numerical experiments, it turns out that strongly imposed Dirichlet boundary conditions lead to ..."
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Cited by 5 (0 self)
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Abstract. We investigate the Continuous Interior Penalty (CIP) stabilization method for higher order nite elements applied to a convection diffusion equation with a small diffusion parameter . Performing numerical experiments, it turns out that strongly imposed Dirichlet boundary conditions lead to relatively bad numerical solutions. However, if the Dirichlet boundary conditions are imposed on the inow part of the boundary in a weak sense and additionally on the whole boundary in an weighted weak sense due to Nitsche then one obtains reasonable numerical results. In many cases, this holds even in the limit case where the parameter of the CIP stabilization is zero, i.e., where the standard Galerkin discretization is applied. We present an analysis which explains this effect. Key words. diffusionconvectionreaction equation, nite elements, Nitsche type boundary conditions, error estimates AMS subject classications. 65N15, 65N30, 65N50 1. Introduction. We
Unified edgeoriented stabilization of nonconforming finite element methods for incompressible flow problems
 J. Numer. Math
"... Summary. This paper deals with various aspects of edgeoriented stabilization techniques for nonconforming finite element methods for the numerical solution of incompressible flow problems. We discuss two separate classes of problems which require appropriate stabilization techniques: First, the lac ..."
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Cited by 4 (3 self)
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Summary. This paper deals with various aspects of edgeoriented stabilization techniques for nonconforming finite element methods for the numerical solution of incompressible flow problems. We discuss two separate classes of problems which require appropriate stabilization techniques: First, the lack of coercivity for nonconforming low order approximations for treating problems with the symmetric deformation tensor instead of the gradient formulation in the momentum equation (‘Korn’s inequality’) which particularly leads to convergence problems of the iterative solvers for small Reynolds (Re) numbers. Second, numerical instabilities for high Re numbers or whenever convective operators are dominant such that the standard Galerkin formulation fails and leads to spurious oscillations. We show that the right choice of edgeoriented stabilization is able to provide simultanously excellent results regarding robustness and accuracy for both seemingly different cases of problems, and we discuss the sensitivity of the involved parameters w.r.t. mesh distortions and variations of the Re number. Moreover, we explain how efficient multigrid solvers can be constructed to circumvent the problems with the arising ‘nonstandard ’ FEM data structures, and we provide several examples for the numerical efficiency for realistic flow configurations with benchmarking character. 1
New robust nonconforming finite elements of higher order
"... We study second order nonconforming finite elements as members of a new family of higher order approaches which behave optimally not only on multilevel refined grids, but also on perturbed grids which are still shape regular but which consist no longer of asymptotically affine equivalent mesh cells. ..."
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Cited by 4 (3 self)
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We study second order nonconforming finite elements as members of a new family of higher order approaches which behave optimally not only on multilevel refined grids, but also on perturbed grids which are still shape regular but which consist no longer of asymptotically affine equivalent mesh cells. We present two approaches to prevent this order reduction: The first one is based on the use of nonparametric basis functions which are defined as polynomials on the original mesh cell. In the second approach, we define all basis functions on the reference element but add one or more nonconforming cell bubble functions which can be removed at the end by static condensation. For the last approach, we prove optimal estimates for the approximation and consistency error and derive optimal estimates for the discretization error in the case of a Poisson problem. Furthermore, we construct and analyze numerically 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 NavierStokes 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 FEMmultigrid combination (together with discontinuous pressure approximations) appear to be very advantageous candidates for realistic flow simulation tools, particularly on (parallel) high performance computing systems.