115 research outputs found
Analysis of Compatible Discrete Operator Schemes for the Stokes Equations on Polyhedral Meshes
Compatible Discrete Operator schemes preserve basic properties of the
continuous model at the discrete level. They combine discrete differential
operators that discretize exactly topological laws and discrete Hodge operators
that approximate constitutive relations. We devise and analyze two families of
such schemes for the Stokes equations in curl formulation, with the pressure
degrees of freedom located at either mesh vertices or cells. The schemes ensure
local mass and momentum conservation. We prove discrete stability by
establishing novel discrete Poincar\'e inequalities. Using commutators related
to the consistency error, we derive error estimates with first-order
convergence rates for smooth solutions. We analyze two strategies for
discretizing the external load, so as to deliver tight error estimates when the
external load has a large irrotational or divergence-free part. Finally,
numerical results are presented on three-dimensional polyhedral meshes
A nonconforming finite element method for the Biot’s consolidation model in poroelasticity
A stable finite element scheme that avoids pressure oscillations for a three-field Biot’s model in poroelasticity is considered. The involved variables are the displacements, fluid flux (Darcy velocity), and the pore pressure, and they are discretized by using the lowest possible approximation order: Crouzeix–Raviart finite elements for the displacements, lowest order Raviart–Thomas-Nédélec elements for the Darcy velocity, and piecewise constant approximation for the pressure. Mass-lumping technique is introduced for the Raviart–Thomas-Nédélec elements in order to eliminate the Darcy velocity and, therefore, reduce the computational cost. We show convergence of the discrete scheme which is implicit in time and use these types of elements in space with and without mass-lumping. Finally, numerical experiments illustrate the convergence of the method and show its effectiveness to avoid spurious pressure oscillations when mass lumping for the Raviart–Thomas-Nédélec elements is used
A low-order nonconforming method for linear elasticity on general meshes
In this work we construct a low-order nonconforming approximation method for
linear elasticity problems supporting general meshes and valid in two and three
space dimensions. The method is obtained by hacking the Hybrid High-Order
method, that requires the use of polynomials of degree for stability.
Specifically, we show that coercivity can be recovered for by introducing
a novel term that penalises the jumps of the displacement reconstruction across
mesh faces. This term plays a key role in the fulfillment of a discrete Korn
inequality on broken polynomial spaces, for which a novel proof valid for
general polyhedral meshes is provided. Locking-free error estimates are derived
for both the energy- and the -norms of the error, that are shown to
convergence, for smooth solutions, as and , respectively (here,
denotes the meshsize). A thorough numerical validation on a complete panel of
two- and three-dimensional test cases is provided.Comment: 26 pages, 6 tables, and 4 Figure
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Advanced Computational Engineering
The finite element method is the established simulation tool for the numerical solution of partial differential equations in many engineering problems with many mathematical developments such as mixed finite element methods (FEMs) and other nonstandard FEMs like least-squares, nonconforming, and discontinuous Galerkin (dG) FEMs. Various aspects on this plus related topics ranging from order-reduction methods to isogeometric analysis has been discussed amongst the pariticpants form mathematics and engineering for a large range of applications
Divergence-preserving reconstructions on polygons and a really pressure-robust virtual element method for the Stokes problem
Non divergence-free discretisations for the incompressible Stokes problem may suffer from a lack of pressure-robustness characterised by large discretisations errors due to irrotational forces in the momentum balance. This paper argues that also divergence-free virtual element methods (VEM) on polygonal meshes are not really pressure-robust as long as the right-hand side is not discretised in a careful manner. To be able to evaluate the right-hand side for the testfunctions, some explicit interpolation of the virtual testfunctions is needed that can be evaluated pointwise everywhere. The standard discretisation via an L2 -bestapproximation does not preserve the divergence and so destroys the orthogonality between divergence-free testfunctions and possibly eminent gradient forces in the right-hand side. To repair this orthogonality and restore pressure-robustness another divergence-preserving reconstruction is suggested based on Raviart--Thomas approximations on local subtriangulations of the polygons. All findings are proven theoretically and are demonstrated numerically in two dimensions. The construction is also interesting for hybrid high-order methods on polygonal or polyhedral meshes
Finite Element Decomposition and Minimal Extension for Flow Equations
In the simulation of flows, the correct treatment of the pressure variable is the key to stable time-integration schemes. This paper contributes a new approach based on the theory of differential-algebraic equations. Motivated by the index reduction technique of minimal extension, a decomposition of finite element spaces is proposed that ensures stable and accurate approximations. The presented decomposition -- for standard finite element spaces used in CFD -- preserves sparsity and does not call on variable transformations which might change the meaning of the variables. Since the method is eventually an index reduction, high index effects leading to instabilities are eliminated. As a result, all constraints are maintained and one can apply semi-explicit time integration schemes
Recovery Techniques For Finite Element Methods And Their Applications
Recovery techniques are important post-processing methods to obtain improved approximate solutions from primary data with reasonable cost. The practical us- age of recovery techniques is not only to improve the quality of approximation, but also to provide an asymptotically exact posteriori error estimators for adaptive meth- ods. This dissertation presents recovery techniques for nonconforming finite element methods and high order derivative as well as applications of gradient recovery.
Our first target is to develop a systematic gradient recovery technique for Crouzeix- Raviart element. The proposed method uses finite element solution to build a better approximation of the exact gradient based on local least square fittings. Due to poly- nomial preserving property of least square fitting, it is easy to show that the new proposed method preserves quadratic polynomials. In addition, the proposed gra- dient recovery is linearly bounded. Numerical tests indicate the recovered gradient is superconvergent to the exact gradient for both second order elliptic equation and Stokes equation. The gradient recovery technique can be used in a posteriori error
estimates for Crouzeix-Raviart element, which is relatively simple to implement and problem independent.
Our second target is to propose and analyze a new effective Hessian recovery for continuous finite element of arbitrary order. The proposed Hessian recovery is based on polynomial preserving recovery. The proposed method preserves polynomials of degree (k + 1) on general unstructured meshes and polynomials of degree (k + 2) on translation invariant meshes. Based on it polynomial preserving property, we can able to prove superconvergence of the proposed method on mildly structured meshes. In addition, we establish the ultraconvergence result for the new Hessian recovery technique on translation invariant finite element space of arbitrary order.
Our third target is to demonstrate application of gradient recovery in eigenvalue computation. We propose two superconvergent two-grid methods for elliptic eigen- value problems by taking advantage of two-gird method, two-space method, shifted- inverse power method, and gradient recovery enhancement. Theoretical and numer- ical results reveal that the proposed methods provide superconvergent eigenfunction approximation and ultraconvergent eigenvalue approximation. In addition, two mul- tilevel adaptive methods based recovery type a posterior error estimate are proposed
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