Piecewise Constant Finite Elements Research Articles

A stable finite element method for low inertia undulatory locomotion in three dimensions

We present and analyse a numerical method for understanding the low-inertia dynamics of an open, inextensible viscoelastic rod - a long and thin three dimensional object - representing the body of a long, thin microswimmer. Our model allows for both elastic and viscous, bending and twisting deformations and describes the evolution of the midline curve of the rod as well as an orthonormal frame which fully determines the rod's three dimensional geometry. The numerical method is based on using a combination of piecewise linear and piecewise constant finite element functions based on a novel rewriting of the model equations. We derive a stability estimate for the semi-discrete scheme and show that at the fully discrete level that we have good control over the length element and preserve the frame orthonormality conditions up to machine precision. Numerical experiments demonstrate both the good properties of the method as well as the applicability of the method for simulating undulatory locomotion in the low-inertia regime.

An accelerated primal-dual iterative scheme for the L2-TV regularized model of linear inverse problems

A model with the and total variational (TV) regularization terms for linear inverse problems is considered. The regularized model is reformulated as a saddle-point problem, and the primal and dual variables are discretized in the piecewise affine and piecewise constant finite element spaces, respectively. An accelerated primal-dual iterative scheme with an convergence rate is proposed for the discretized problem, where is the iteration counter. Both the regularization and perturbation errors of the regularized model, and the finite element discretization and iteration errors of the accelerated primal-dual scheme, are estimated. Preliminary numerical results are reported to show the efficiency of the proposed iterative scheme.

An Augmented Lagrangian Preconditioner for the 3D Stationary Incompressible Navier--Stokes Equations at High Reynolds Number

In Benzi & Olshanskii (SIAM J.~Sci.~Comput., 28(6) (2006)) a preconditioner of augmented Lagrangian type was presented for the two-dimensional stationary incompressible Navier--Stokes equations that exhibits convergence almost independent of Reynolds number. The algorithm relies on a highly specialized multigrid method involving a custom prolongation operator and is tightly coupled to the use of piecewise constant finite elements for the pressure. However, the prolongation operator and velocity element used do not directly extend to three dimensions: the local solves necessary in the prolongation operator do not satisfy the inf-sup condition. In this work we generalize the preconditioner to three dimensions, proposing alternative finite elements for the velocity and prolongation operators for which the preconditioner works robustly. The solver is effective at high Reynolds number: on a three-dimensional lid-driven cavity problem with approximately one billion degrees of freedom, the average number of Krylov iterations per Newton step varies from $7$ at $\mathrm{Re} = 4.5$ to $3$ at $\mathrm{Re} = 1000$ and $5$ at $\mathrm{Re} = 5000$.

A projection‐based time‐splitting algorithm for approximating nematic liquid crystal flows with stretching

AbstractA numerical method is developed for solving a system of partial differential equations modeling the flow of a nematic liquid crystal fluid with stretching effect, which takes into account the geometrical shape of its molecules. This system couples the velocity vector, the scalar pressure and the director vector representing the direction along which the molecules are oriented. The scheme is designed by using finite elements in space and a time‐splitting algorithm to uncouple the calculation of the variables: the velocity and pressure are computed by using a projection‐based algorithm and the director is computed jointly to an auxiliary variable. Moreover, the computation of this auxiliary variable can be avoided at the discrete level by using piecewise constant finite elements in its approximation. Finally, we use a pressure stabilization technique allowing a stable equal‐order interpolation for the velocity and the pressure. Numerical experiments concerning annihilation of singularities are presented to show the stability and efficiency of the scheme.

Adaptive techniques in SOLD methods

We present new results where free parameters in spurious oscillations at layers diminishing (SOLD) method are adaptively chosen. Provided numerical results are for conforming piecewise linear finite element space with free parameters from piecewise constant finite element space. We focus on a higher order convergence we discovered in previous paper by Lukáš (2015).

A generalized multiscale finite element method for the Brinkman equation

In this paper we consider the numerical upscaling of the Brinkman equation in the presence of high-contrast permeability fields. We develop and analyze a robust and efficient Generalized Multiscale Finite Element Method (GMsFEM) for the Brinkman model in two dimensions. In the fine grid, we use mixed finite element method with the velocity and pressure being continuous piecewise quadratic and piecewise constant finite element spaces, respectively. Using the GMsFEM framework we construct suitable coarse-scale spaces for the velocity and pressure that yield a robust mixed GMsFEM. We develop a novel approach to construct a coarse approximation for the velocity snapshot space and a robust small offline space for the velocity space. The stability of the mixed GMsFEM and a priori error estimates are derived. A variety of two-dimensional numerical examples are presented to illustrate the effectiveness of the algorithm.

Locking-free finite element method for a bending moment formulation of Timoshenko beams

In this paper we study a finite element formulation for Timoshenko beams. It is known that standard finite elements applied to this model lead to wrong results when the thickness of the beam is small. Here, we consider a mixed formulation in terms of the transverse displacement, rotation, shear stress and bending moment. By using the classical Babuška–Brezzi theory, it is proved that the resulting variational formulation is well posed. We discretize it by continuous piecewise linear finite elements for the shear stress and bending moment, and discontinuous piecewise constant finite elements for the displacement and rotation. We prove an optimal (linear) order of convergence in terms of the mesh size for the natural norms and a double order (quadratic) in L2-norms for the shear stress and the bending moment. These estimates involve constants and norms of the solution that are proved to be bounded independently of the beam thickness, which ensures the locking-free character of the method. Numerical tests are reported in order to support our theoretical results.

Asymptotically Compatible Schemes and Applications to Robust Discretization of Nonlocal Models

Many problems in nature, being characterized by a parameter, are of interest both with a fixed parameter value and with the parameter approaching an asymptotic limit. Numerical schemes that are convergent in both regimes offer robust discretizations, which can be highly desirable in practice. The asymptotically compatible schemes studied in this paper meet such objectives for a class of parametrized problems. An abstract mathematical framework is established rigorously here together with applications to the numerical solution of both nonlocal models and their local limits. In particular, the framework can be applied to nonlocal diffusion models and a general state-based peridynamic system parametrized by the horizon radius. Recent findings have exposed the risks associated with some discretizations of nonlocal models when the horizon radius is proportional to the discretization parameter. Thus, it is desirable to develop asymptotically compatible schemes for such models so as to offer robust numerical discretizations to problems involving nonlocal interactions on multiple scales. This work provides new insight in this regard through a careful analysis of related conforming finite element discretizations and the finding is valid under minimal regularity assumptions on exact solutions. It reveals that as long as the finite element space contains continuous piecewise linear functions, the Galerkin finite element approximation is always asymptotically compatible. For piecewise constant finite element, whenever applicable, it is shown that a correct local limit solution can also be obtained as long as the discretization (mesh) parameter decreases faster than the modeling (horizon) parameter does. These results can be used to guide future computational studies of nonlocal problems.

A mixed finite element method for nearly incompressible elasticity and Stokes equations using primal and dual meshes with quadrilateral and hexahedral grids

We consider a mixed finite element method for approximating the solution of nearly incompressible elasticity and Stokes equations. The finite element method is based on quadrilateral and hexahedral triangulation using primal and dual meshes. We use the standard bilinear and trilinear finite element space enriched with element-wise defined bubble functions with respect to the primal mesh for the displacement or velocity, whereas the pressure space is discretized by using a piecewise constant finite element space with respect to the dual mesh.

An augmented mixed finite element method for the vorticity–velocity–pressure formulation of the Stokes equations

This paper deals with the numerical approximation of the stationary two-dimensional Stokes equations, formulated in terms of vorticity, velocity and pressure, with non-standard boundary conditions. Here, by introducing a Galerkin least-squares term, we end up with a stabilized variational formulation that can be recast as a twofold saddle point problem. We propose two families of mixed finite elements to solve the discrete problem, in the first family, the unknowns are approximated by piecewise continuous and quadratic elements, Brezzi–Douglas–Marini, and piecewise constant finite elements, respectively, while in the second family, the unknowns are approximated by piecewise linear and continuous, Raviart–Thomas, and piecewise constant finite elements, respectively. The wellposedness of the resulting continuous and discrete variational problems are studied employing an extension of the Babuška–Brezzi theory. We establish a priori error estimates in the natural norms, and we finally report some numerical experiments illustrating the behavior of the proposed schemes and confirming our theoretical findings on structured and unstructured meshes. Additional examples of cases not covered by our theory are also presented.

Total Variation Minimization with Finite Elements: Convergence and Iterative Solution

The numerical solution of a convex minimization problem involving the nonsmooth total variation norm is analyzed. Consistent finite element discretizations that avoid regularizations lead to simple convergence proofs in the case of piecewise affine, globally continuous finite elements. For the approximation with piecewise constant finite elements it is proved that convergence to the exact solution cannot be expected in general. The iterative solution is based on a regularized $L^2$ flow of the energy functional, and convergence of the iteration to a stationary point is proved under a moderate constraint on the time-step size. The extension of the techniques to an energy functional that involves a negative order term is discussed. Numerical experiments that illustrate the theoretical results are presented.

Negative norm error control for second-kind convolution Volterra equations

Summary. We consider a piecewise constant finite element approximation to the convolution Volterra equation problem of the second kind: find $u$ such that $u = f + \phi * u$ in a time interval $[0,T]$ . An a posteriori estimate of the error measured in the $W^{-1}_p(0,T)$ norm is developed and used to provide a time step selection criterion for an adaptive solution algorithm. Numerical examples are given for problems in which $\phi$ is of a form typical in viscoelasticity theory.

Error estimates for a discretized Galerkin method for a boundary integral equation in two dimensions

AbstractWe present a priori and a posteriori estimates for the error between the Galerkin and a discretized Galerkin method for the boundary integral equation for the single layer potential on the square plate. Using piecewise constant finite elements on a rectangular mesh we study the error coming from numerical integration. The crucial point of our analysis is the estimation of some error constants, and we demonstrate that this is necessary if our methods are to be used. After the determination of these constants we are in the position to prove invertibility and quasioptimal convergence results for our numerical scheme, if the chosen numerical integration formulas are sufficiently precise. © 1992 John Wiley & Sons, Inc.

A Spectral General Circulation Model Using a Piecewise-Constant Finite-Element Representation on a Hybrid Vertical Coordinate System

Abstract A numerical scheme for the vertical discretization of primitive equations in a generalized pressure-type coordinate is developed through application of the Galerkin formalism with piecewise-constant finite elements: this methodology affords an elegant—and direct—mean of formulating conservative discretization schemes without the arbitrariness that usually characterizes the development of finite differences. The form of the resulting semidiscrete equations is equivalent to some second-order accurate finite-difference approximation to the continuous equations. Flexibility of this scheme in the choice of different layers for projecting the thermodynamic and momentum variables effectively allows for staggering of these variables in the vertical. Numerical integrations performed with this scheme at various vertical resolutions have revealed the sensitivity of the simulated circulation to resolution in the lower stratosphere. We found that application of the “lid” upper boundary condition at a finite h...

Implementation of the discontinuous finite element method for hyperbolic equations

We present in this paper a method to implement the discontinuous piecewise constant finite element method for linear hyperbolic equations. A particular emphasis is made on an algorithm of element reordering allowing an element-by-element computation of the solution. Practical order of convergence of the method is calculated on a test problem giving a better accuracy than the expected theoretical order even for nonuniform triangular meshes.

Experimental results on the accuracy of a global forecast spectral model with different vertical discretization schemes

Abstract A new vertical discretization scheme has been designed and tested in a global forecast spectral model. The scheme uses piecewise constant finite elements and guarantees conservation of global properties such as total mass, energy and angular momentum. A series of 7‐day global forecasts have been performed and compared with the corresponding forecasts obtained from the same global spectral model with a vertical staggered finite‐difference method and another with an unstaggered linear finite‐element method. The results suggest that, on the average, the new scheme with conservative properties performs better than the finite‐difference scheme but gives poorer results in the long‐wave amplitude and phase speed than the linear finite‐element scheme. The major difference between the schemes always appears near the bottom and the top boundaries which seems to be the main source of error in the model.

Full discretization of the porous medium/fast diffusion equation based on its very weak formulation

SKA ‡ Abstract. The very weak formulation of the porous medium/fast diffusion equation yields an evolution problem in a Gelfand triple with the pivot space H 1. This allows to employ methods of the theory of monotone operators in order to study fully discrete approximations combining a Galerkin method (including conforming finite element methods) with the backward Euler scheme. Convergence is shown even for rough initial data and right-hand sides. The theoretical results are illustrated for the piecewise constant finite element approximation of the porous medium equation with the�-distribution as initial value. As a byproduct, L p -stability of theH 1 -orthogonal projection onto the space of piecewise constant functions is shown. ut − �(juj p−2 u) = f; p >1; u= u(x;t); u(�;0) = u0; (x;t) 2 � (0;T); whereR d is a nice bounded domain, −� is the Laplace operator acting on the spatial variables, u0 are given initial data, f is a given right-hand side and we assume some boundary condition on the boundary of . We immediately see that for p = 2 this is the classical heat equation. For p > 2, it is referred to as the porous medium equation. In this case u is the density of the gas at a given point and time. For 1 2, in the one dimensional case, the solution is given by (5.2). We will later use this in numerical experiments. The solution is also known in higher dimensions, see again Vazquez (22, Chapter 4). Throughout the article, we focus on the following generalization of the above