arXiv subjects
Publications and source records attributed to Philip L. Lederer.
We consider mixed finite element methods with exact symmetric stress tensors. We derive a new quasi-optimal a priori error estimate uniformly valid with respect to the compressibility. For the a posteriori error analysis we consider the Prager-Synge hypercircle principle and introduce a new estimate uniformly valid in the incompressible limit. All estimates are validated by numerical examples.
We are studying the efficient solution of the system of linear equation stemming from the mass conserving mixed stress (MCS) method discretization of the Stokes equations. To that end we perform static condensation to arrive at a system for the pressure and velocity unknowns. An auxiliary space preconditioner for the positive definite velocity block makes use of efficient and scalable solvers for conforming Finite Element spaces of low order and is analyzed with emphasis placed on the polynomial degree of the discretization. Numerical experiments demonstrate the potential of this approach and the efficiency of the implementation.
We consider mixed finite element approximations of viscous, plastic Bingham flow in a cylindrical pipe. A novel a priori and a posteriori error analysis is introduced which is based on a discrete mesh dependent norm for the normalized Lagrange multiplier. This allows proving stability for various conforming finite elements. Numerical examples are presented to support the theory and to demonstrate adaptive mesh refinement.
The quasi-static multiple network poroelastic theory (MPET) model, first introduced in the context of geomechanics, has recently found new applications in medicine. In practice, the parameters in the MPET equations can vary over several orders of magnitude which makes their stable discretization and fast solution a challenging task. Here, a new efficient parameter-robust hybridized discontinuous Galerkin method, which also features fluid mass conservation, is proposed for the MPET model. Its stability analysis which is crucial for the well-posedness of the discrete problem is performed and cost-efficient fast parameter-robust preconditioners are derived. We present a series of numerical computations for a 4-network MPET model of a human brain which support the performance of the new algorithms.
We derive optimal and asymptotically exact a posteriori error estimates for the approximation of the Laplace eigenvalue problem. To do so, we combine two results from the literature. First, we use the hypercircle techniques developed for mixed eigenvalue approximations with Raviart-Thomas Finite elements. In addition, we use the post-processings introduced for the eigenvalue and eigenfunction based on mixed approximations with the Brezzi-Douglas-Marini Finite element. To combine these approaches, we define a novel additional local post-processing for the fluxes that appropriately modifies the divergence. Consequently, the new flux can be used to derive upper bounds and still shows good approximation properties. Numerical examples validate the theory and motivate the use of an adaptive mesh refinement.
This paper aims to improve guaranteed error control for the Stokes problem with a focus on pressure-robustness, i.e. for discretisations that compute a discrete velocity that is independent of the exact pressure. A Prager--Synge type result relates the velocity errors of divergence-free primal and perfectly equilibrated dual mixed methods for the velocity stress. The first main result of the paper is a framework with relaxed constraints on the primal and dual method. This enables to use a recently developed mass conserving mixed stress discretisation for the design of equilibrated fluxes and to obtain pressure-independent guaranteed upper bounds for any pressure-robust (not necessarily divergence-free) primal discretisation. The second main result is a provably efficient local design of the equilibrated fluxes with comparably low numerical costs. Numerical examples verify the theoretical findings and show that efficiency indices of our novel guaranteed upper bounds are close to one.
We consider the quasi-static Biot's consolidation model in a three-field formulation with the three unknown physical quantities of interest being the displacement $\boldsymbol{u}$ of the solid matrix, the seepage velocity $\boldsymbol{v}$ of the fluid and the pore pressure $p$. As conservation of fluid mass is a leading physical principle in poromechanics, we preserve this property using an $\boldsymbol{H}(\operatorname{div})$-conforming ansatz for $\boldsymbol{u}$ and $\boldsymbol{v}$ together with an appropriate pressure space. This results in Stokes and Darcy stability and exact, that is, pointwise mass conservation of the discrete model. The proposed discretization technique combines a hybridized discontinuous Galerkin method for the elasticity subproblem with a mixed method for the flow subproblem, also handled by hybridization. The latter allows for a static condensation step to eliminate the seepage velocity from the system while preserving mass conservation. The system to be solved finally only contains degrees of freedom related to $\boldsymbol{u}$ and $p$ resulting from the hybridization process and thus provides, especially for higher-order approximations, a very cost-efficient family of physics-oriented space discretizations for poroelasticity problems. We present the construction of the discrete model, theoretical results related to its uniform well-posedness along with optimal error estimates and parameter-robust preconditioners as a key tool for developing uniformly convergent iterative solvers. Finally, the cost-efficiency of the proposed approach is illustrated in a series of numerical tests for three-dimensional test cases.
We introduce a new discretization for the stream function formulation of the incompressible Stokes equations in two and three space dimensions. The method is strongly related to the Hellan-Herrmann-Johnson method and is based on the recently discovered mass conserving mixed stress formulation [J. Gopalakrishnan, P.L. Lederer, J. Schöberl, IMA Journal of numerical Analysis, 2019] that approximates the velocity in an $H(\operatorname{div})$-conforming space and introduces a new stress-like variable for the approximation of the gradient of the velocity within the function space $H(\operatorname{curl}\operatorname{div})$. The properties of the (discrete) de Rham complex allows to extend this method to a stream function formulation in two and three space dimensions. We present a detailed stability analysis in the continuous and the discrete setting where the stream function $ψ$ and its approximation $ψ_h$ are elements of $H(\operatorname{curl})$ and the $H(\operatorname{curl})$-conforming Nédélec finite element space, respectively. We conclude with an error analysis revealing optimal convergence rates for the error of the discrete velocity $u_h = \operatorname{curl}(ψ_h)$ measured in a discrete $H^1$-norm. We present numerical examples to validate our findings and discuss structure-preserving properties such as pressure-robustness.
The embedded discontinuous Galerkin (EDG) finite element method for the Stokes problem results in a point-wise divergence-free approximate velocity on cells. However, the approximate velocity is not H(div)-conforming and it can be shown that this is the reason that the EDG method is not pressure-robust, i.e., the error in the velocity depends on the continuous pressure. In this paper we present a local reconstruction operator that maps discretely divergence-free test functions to exactly divergence-free test functions. This local reconstruction operator restores pressure-robustness by only changing the right hand side of the discretization, similar to the reconstruction operator recently introduced for the Taylor--Hood and mini elements by Lederer et al. (SIAM J. Numer. Anal., 55 (2017), pp. 1291--1314). We present an a priori error analysis of the discretization showing optimal convergence rates and pressure-robustness of the velocity error. These results are verified by numerical examples. The motivation for this research is that the resulting EDG method combines the versatility of discontinuous Galerkin methods with the computational efficiency of continuous Galerkin methods and accuracy of pressure-robust finite element methods.
In this work we consider the numerical solution of incompressible flows on two-dimensional manifolds. Whereas the compatibility demands of the velocity and the pressure spaces are known from the flat case one further has to deal with the approximation of a velocity field that lies only in the tangential space of the given geometry. Abandoning $H^1$-conformity allows us to construct finite elements which are -- due to an application of the Piola transformation -- exactly tangential. To reintroduce continuity (in a weak sense) we make use of (hybrid) discontinuous Galerkin techniques. To further improve this approach, $H(\operatorname{div}_Γ)$-conforming finite elements can be used to obtain exactly divergence-free velocity solutions. We present several new finite element discretizations. On a number of numerical examples we examine and compare their qualitative properties and accuracy.
We introduce a new discretization of a mixed formulation of the incompressible Stokes equations that includes symmetric viscous stresses. The method is built upon a mass conserving mixed formulation that we recently studied. The improvement in this work is a new method that directly approximates the viscous fluid stress $σ$, enforcing its symmetry weakly. The finite element space in which the stress is approximated consists of matrix-valued functions having continuous "normal-tangential" components across element interfaces. Stability is achieved by adding certain matrix bubbles that were introduced earlier in the literature on finite elements for linear elasticity. Like the earlier work, the new method here approximates the fluid velocity $u$ using $H(\operatorname{div})$-conforming finite elements, thus providing exact mass conservation. Our error analysis shows optimal convergence rates for the pressure and the stress variables. An additional post processing yields an optimally convergent velocity satisfying exact mass conservation. The method is also pressure robust.
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.
We propose a new discretization of a mixed stress formulation of the Stokes equations. The velocity $u$ is approximated with $H(\operatorname{div})$-conforming finite elements providing exact mass conservation. While many standard methods use $H^1$-conforming spaces for the discrete velocity, $H(\operatorname{div})$-conformity fits the considered variational formulation in this work. A new stress-like variable $σ$ equalling the gradient of the velocity is set within a new function space $H(\operatorname{curl} \operatorname{div})$. New matrix-valued finite elements having continuous "normal-tangential" components are constructed to approximate functions in $H(\operatorname{curl} \operatorname{div})$. An error analysis concludes with optimal rates of convergence for errors in $u$ (measured in a discrete $H^1$-norm), errors in $σ$ (measured in $L^2$) and the pressure $p$ (also measured in $L^2$). The exact mass conservation property is directly related to another structure-preservation property called pressure robustness, as shown by pressure-independent velocity error estimates. The computational cost measured in terms of interface degrees of freedom is comparable to old and new Stokes discretizations.
The present work is the second part of a pair of papers, considering Hybrid Discontinuous Galerkin methods with relaxed H(div)-conformity. The first part mainly dealt with presenting a robust analysis with respect to the mesh size $h$ and the introduction of a reconstruction operator to restore divergence-conformity and pressure robustness using a non conforming right hand side. The aim of this part is the presentation of a high order polynomial robust analysis for the relaxed $H(\operatorname{div})$-conforming Hybrid Discontinuous Galerkin discretization of the two dimensional Stokes problem. It is based on the recently proven polynomial robust LBB-condition for BDM elements [P. L. Lederer, J. Schöberl, IMA Journal of Numerical Analysis, 2017] and is derived by a direct approach instead of using a best approximation Céa like result. We further treat the impact of the reconstruction operator on the $hp$ analysis and present a numerical investigation considering polynomial robustness. We conclude the paper presenting an efficient operator splitting time integration scheme for the Navier--Stokes equations which is based on the methods recently presented in [C. Lehrenfeld, J. Schöberl, \emph{Comp. Meth. Appl. Mech. Eng.}, 361 (2016)] and includes the ideas of the reconstruction operator.
We propose a new discretization method for the Stokes equations. The method is an improved version of the method recently presented in [C. Lehrenfeld, J. Schöberl, Comp. Meth. Appl. Mech. Eng., 361 (2016)] which is based on an $H(\operatorname{div})$-conforming finite element space and a Hybrid Discontinuous Galerkin (HDG) formulation of the viscous forces. $H(\operatorname{div})$-conformity results in favourable properties such as pointwise divergence free solutions and pressure-robustness. However, for the approximation of the velocity with a polynomial degree $k$ it requires unknowns of degree $k$ on every facet of the mesh. In view of the superconvergence property of other HDG methods, where only unknowns of polynomial degree $k-1$ on the facets are required to obtain an accurate polynomial approximation of order $k$ (possibly after a local post-processing) this is sub-optimal. The key idea in this paper is to slightly relax the $H(\operatorname{div})$-conformity so that only unknowns of polynomial degree $k-1$ are involved for normal-continuity. This allows for optimality of the method also in the sense of superconvergent HDG methods. In order not to loose the benefits of $H(\operatorname{div})$-conformity we introduce a cheap reconstruction operator which restores pressure-robustness and pointwise divergence free solutions and suits well to the finite element space with relaxed $H(\operatorname{div})$-conformity. We present this new method, carry out a thorough $h$-version error analysis and demonstrate the performance of the method on numerical examples.
In this work we consider a discontinuous Galerkin method for the discretization of the Stokes problem. We use $H(\textrm{div})$-conforming finite elements as they provide major benefits such as exact mass conservation and pressure-independent error estimates. The main aspect of this work lies in the analysis of high order approximations. We show that the considered method is uniformly stable with respect to the polynomial order $k$ and provides optimal error estimates $ \| \boldsymbol{u} - \boldsymbol{u}_h \|_{1_h} + \| Π^{Q_h}p-p_h \| \le c \left( h/k \right)^s \| \boldsymbol{u} \|_{s+1} $. To derive those estimates, we prove a $k$-robust LBB condition. This proof is based on a polynomial $H^2$-stable extension operator. This extension operator itself is of interest for the numerical analysis of $C^0$-continuous discontinuous Galerkin methods for $4^{th}$ order problems.
Classical inf-sup stable mixed finite elements for the incompressible (Navier-)Stokes equations are not pressure-robust, i.e., their velocity errors depend on the continuous pressure. However, a modification only in the right hand side of a Stokes discretization is able to reestablish pressure-robustness, as shown recently for several inf-sup stable Stokes elements with discontinuous discrete pressures. In this contribution, this idea is extended to low and high order Taylor-Hood and mini elements, which have continuous discrete pressures. For the modification of the right hand side a velocity reconstruction operator is constructed that maps discretely divergence-free test functions to exactly divergence-free ones. The reconstruction is based on local $H(\mathrm{div})$-conforming flux equilibration on vertex patches, and fulfills certain orthogonality properties to provide consistency and optimal a-priori error estimates. Numerical examples for the incompressible Stokes and Navier-Stokes equations confirm that the new pressure-robust Taylor-Hood and mini elements converge with optimal order and outperform significantly the classical versions of those elements when the continuous pressure is comparably large.