Linear Finite Element Approximation Research Articles

Optimal convergence analysis of the unilateral contact problem with and without Tresca friction conditions by the penalty method

We study the linear finite element approximation of the elasticity equations with and without unilateral friction contact (of Tresca type) conditions in a polygonal or polyhedral domain. The unilateral contact condition is weakly imposed by the penalty method. We derive error estimates which depend on the penalty parameter ε and the mesh size h . In fact, under the H 3 2 + ν ( Ω ) , 0 < ν ≤ 1 2 , regularity of the solution of the contact problems (with and without friction) and with the requirement ε > h , we prove a convergence rate of O ( h 1 2 + ν + ε 1 2 + ν ) in the energy norm. Therefore, if the penalty parameter is taken as ε : = c h θ where 0 < θ ≤ 1 , the convergence rate of O ( h θ ( 1 2 + ν ) ) is obtained. In particular, we obtain an optimal linear convergence when ε behaves like h (i.e. θ = 1 ) and ν = 1 2 .

Fully Discrete Finite Element Methods for Two-Dimensional Bingham Flows

This paper presents fully discrete stabilized finite element methods for two-dimensional Bingham fluid flow based on the method of regularization. Motivated by the Brezzi-Pitkäranta stabilized finite element method, the equal-order piecewise linear finite element approximation is used for both the velocity and the pressure. Based on Euler semi-implicit scheme, a fully discrete scheme is introduced. It is shown that the proposed fully discrete stabilized finite element scheme results in the h1/2 error order for the velocity in the discrete norms corresponding to L2(0,T;H1(Ω)2)∩L∞(0,T;L2(Ω)2).

Domain decomposition algorithms for problems of thermomechanical contact between several elastic bodies

We consider a thermoelastic multibody contact problem for finite bodies with unilateral mechanical and imperfect thermal contact conditions. Using a penalty method, we obtain a weak formulation of this problem in the form of a system of linear and nonlinear variational equations in Hilbert space. To solve this variational system, we propose a class of iterative Robin type domain decomposition algorithms. In each iterative step of these algorithms one have to solve two linear variational equations for each of the bodies, which correspond to heat conduction problem with Newton boundary conditions on the possible contact areas and linear elasticity problem with additional volume forces and Robin boundary conditions respectively. The program implementation of proposed algorithms is made for plane thermoelastic contact problems with the use of linear and quadratic finite element approximations on triangles. The numerical analysis is performed for one-body and two-body thermoelastic contact problems.

A note on a priori $$\mathbf {L^p}$$ L p -error estimates for the obstacle problem

This paper is concerned with a priori error estimates for the piecewise linear finite element approximation of the classical obstacle problem. We demonstrate by means of two one-dimensional counterexamples that the \(L^2\)-error between the exact solution u and the finite element approximation \(u_h\) is typically not of order two even if the exact solution is in \(H^2(\varOmega )\) and an estimate of the form \(\Vert u - u_h\Vert _{H^1} \le {Ch}\) holds true. This shows that the classical Aubin–Nitsche trick which yields a doubling of the order of convergence when passing over from the \(H^1\)- to the \(L^2\)-norm cannot be generalized to the obstacle problem.

Analysis of the Finite Element Method for the Laplace–Beltrami Equation on Surfaces with Regions of High Curvature Using Graded Meshes

We derive error estimates for the piecewise linear finite element approximation of the Laplace–Beltrami operator on a bounded, orientable, $$C^3$$ , surface without boundary on general shape regular meshes. As an application, we consider a problem where the domain is split into two regions: one which has relatively high curvature and one that has low curvature. Using a graded mesh we prove error estimates that do not depend on the curvature on the high curvature region. Numerical experiments are provided.

$$\mathbf {L^\infty }$$ L ∞ -error estimates for the obstacle problem revisited

In this paper, we present an alternative approach to a priori \(L^\infty \)-error estimates for the piecewise linear finite element approximation of the classical obstacle problem. Our approach is based on stability results for discretized obstacle problems and on error estimates for the finite element approximation of functions under pointwise inequality constraints. As an outcome, we obtain the same order of convergence proven in several works before. In contrast to prior results, our estimates can, for example, also be used to study the situation where the function space is discretized but the obstacle is not modified at all.

Second order approximations for kinetic and potential energies in Maxwell's wave equations

In this paper we propose a numerical scheme for wave type equations with damping and space variable coefficients. Relevant equations of this kind arise for instance in the context of Maxwell's equations, namely, the electric potential equation and the electric field equation. The main motivation to study such class of equations is the crucial role played by the electric potential or the electric field in enhanced drug delivery applications. Our numerical method is based on piecewise linear finite element approximation and it can be regarded as a finite difference method based on non-uniform partitions of the spatial domain. We show that the proposed method leads to second order convergence, in time and space, for the kinetic and potential energies with respect to a discrete L2-norm.

Mesh Quality and More Detailed Error Estimates of Finite Element Method

AbstractIn this paper, we study the role of mesh quality on the accuracy of linear finite element approximation. We derive a more detailed error estimate, which shows explicitly how the shape and size of elements, and symmetry structure of mesh effect on the error of numerical approximation. Two computable parameters Ge and Gv are given to depict the cell geometry property and symmetry structure of the mesh. In compare with the standard a priori error estimates, which only yield information on the asymptotic error behaviour in a global sense, our proposed error estimate considers the effect of local element geometry properties, and is thus more accurate. Under certain conditions, the traditional error estimates and supercovergence results can be derived from the proposed error estimate. Moreover, the estimators Ge and Gv are computable and thus can be used for predicting the variation of errors. Numerical tests are presented to illustrate the performance of the proposed parameters Ge and Gv.

Gradient‐based nodal limiters for artificial diffusion operators in finite element schemes for transport equations

SummaryThis paper presents new linearity‐preserving nodal limiters for enforcing discrete maximum principles in continuous (linear or bilinear) finite element approximations to transport problems with steep fronts. In the process of algebraic flux correction, the oscillatory antidiffusive part of a high‐order base discretization is decomposed into a set of internodal fluxes and constrained to be local extremum dim inishing. The proposed nodal limiter functions are designed to be continuous and satisfy the principle of linearity preservation that implies the preservation of second‐order accuracy in smooth regions. The use of limited nodal gradients makes it possible to circumvent angle conditions and guarantee that the discrete maximum principle holds on arbitrary meshes. A numerical study is performed for linear convection and anisotropic diffusion problems on uniform and distorted meshes in two space dimensions. Copyright © 2017 John Wiley &amp; Sons, Ltd.

Domain decomposition preconditioning for high-frequency Helmholtz problems with absorption

In this paper we give new results on domain decomposition preconditioners for GMRES when computing piecewise linear finite-element approximations of the Helmholtz equation $-\Delta u - (k^2+ \mathrm {i} \varepsilon )u = f$, with absorption parameter $\varepsilon \in \mathbb {R}$. Multigrid approximations of this equation with $\varepsilon \not = 0$ are commonly used as preconditioners for the pure Helmholtz case ($\varepsilon = 0$). However a rigorous theory for such (so-called “shifted Laplace”) preconditioners, either for the pure Helmholtz equation, or even the absorptive equation ($\varepsilon \not =0$), is still missing. We present a new theory for the absorptive equation that provides rates of convergence for (left- or right-) preconditioned GMRES, via estimates of the norm and field of values of the preconditioned matrix. This theory uses a $k$- and $\varepsilon$-explicit coercivity result for the underlying sesquilinear form and shows, for example, that if $|\varepsilon |\sim k^2$, then classical overlapping Additive Schwarz will perform optimally for the damped problem, provided the subdomain and coarse mesh diameters are carefully chosen. Extensive numerical experiments are given that support the theoretical results. While the theory applies to a certain weighted variant of GMRES, the experiments for both weighted and classical GMRES give comparable results. The theory for the absorptive case gives insight into how its domain decomposition approximations perform as preconditioners for the pure Helmholtz case $\varepsilon = 0$. At the end of the paper we propose a (scalable) multilevel preconditioner for the pure Helmholtz problem that has an empirical computation time complexity of about $\mathcal {O}(n^{4/3})$ for solving finite-element systems of size $n=\mathcal {O}(k^3)$, where we have chosen the mesh diameter $h \sim k^{-3/2}$ to avoid the pollution effect. Experiments on problems with $h\sim k^{-1}$, i.e., a fixed number of grid points per wavelength, are also given.

A new equilibrated residual method improving accuracy and efficiency of flux-free error estimates

This paper presents a new methodology to compute guaranteed upper bounds for the energy norm of the error in the context of linear finite element approximations of the reaction–diffusion equation. The new approach revisits the ideas in Parés et al. (2009) [6, 4], with the goal of substantially reducing the computational cost of the flux-free method while retaining the good quality of the bounds. The new methodology provides also a technique to compute equilibrated boundary tractions improving the quality of standard equilibration strategies. The zeroth-order equilibration conditions are imposed using an alternative less restrictive form of the first-order equilibration conditions, along with a new efficient minimization criterion. This new equilibration strategy provides much more accurate upper bounds for the energy and requires only doubling the dimension of the local linear systems of equations to be solved.

An accurate local average contact method for nonmatching meshes

The present paper deals with linear and quadratic finite element approximations of the two and three-dimensional unilateral contact problems between two elastic bodies with nonmatching meshes. We propose a simple noninterpenetration condition on the displacements which is local as the well known node-to-segment and node-to-face conditions and accurate like the mortar approach. This condition consists of averaging locally on a few elements the noninterpenetration. We prove optimal convergence rates in 2D and 3D using various linear and quadratic elements. The Taylor patch test and the Hertzian contact test illustrate the theoretical results and show the capabilities of the method.

Improved ZZ a posteriori error estimators for diffusion problems: Conforming linear elements

In Cai and Zhang (2009), we introduced and analyzed an improved Zienkiewicz–Zhu (ZZ) estimator for the conforming linear finite element approximation to elliptic interface problems. The estimator is based on the piecewise “constant” flux recovery in the H(div;Ω) conforming finite element space. This paper extends the results of Cai and Zhang (2009) to diffusion problems with full diffusion tensor and to the flux recovery both in piecewise constant and piecewise linear H(div) space.

Anisotropic norm-oriented mesh adaptation for a Poisson problem

We present a novel formulation for the mesh adaptation of the approximation of a Partial Differential Equation (PDE). The discussion is restricted to a Poisson problem. The proposed norm-oriented formulation extends the goal-oriented formulation since it is equation-based and uses an adjoint. At the same time, the norm-oriented formulation somewhat supersedes the goal-oriented one since it is basically a solution-convergent method. Indeed, goal-oriented methods rely on the reduction of the error in evaluating a chosen scalar output with the consequence that, as mesh size is increased (more degrees of freedom), only this output is proven to tend to its continuous analog while the solution field itself may not converge. A remarkable quality of goal-oriented metric-based adaptation is the mathematical formulation of the mesh adaptation problem under the form of the optimization, in the well-identified set of metrics, of a well-defined functional. In the new proposed formulation, we amplify this advantage. We search, in the same well-identified set of metrics, the minimum of a norm of the approximation error. The norm is prescribed by the user and the method allows addressing the case of multi-objective adaptation like, for example in aerodynamics, adaptating the mesh for drag, lift and moment in one shot. In this work, we consider the basic linear finite-element approximation and restrict our study to L2 norm in order to enjoy second-order convergence. Numerical examples for the Poisson problem are computed.

Characterization of the Existence of an Enriched Linear Finite Element Approximation Using Biorthogonal Systems

The present paper is intended to give a characterization of the existence of an enriched linear finite element approximation based on biorthogonal systems. It is shown that the enriched element exists if and only if a certain multivariate generalized trapezoidal type cubature formula has a nonzero approximation error. Furthermore, for such an enriched element we derive simple explicit formulas for the basis functions, and show that the approximation error can be written as the sum of the error of the (non-enriched) element plus a perturbation that depends on the enrichment function. Finally, we estimate the approximation error in L 2 norm. We also give an alternative approach to estimate the approximation error, which relies on an appropriate use of the Poincare inequality.

Solving the Stokes Problem on a Massively Parallel Computer

Abstract We describe a numerical procedure for solving the stationary two‐dimensional Stokes problem based on piecewise linear finite element approximations for both velocity and pressure, a regularization technique for stability, and a defect‐correction technique for improving accuracy. Eliminating the velocity unknowns from the algebraic system yields a symmetric positive semidefinite system for pressure which is solved by an inner‐outer iteration. The outer iterations consist of the unpreconditioned conjugate gradient method. The inner iterations, each of which corresponds to solving an elliptic boundary value problem for each velocity component, are solved by the conjugate gradient method with a preconditioning based on the algebraic multi‐level iteration (AMLI) technique. The velocity is found from the computed pressure. The method is optimal in the sense that the computational work is proportional to the number of unknowns. Further, it is designed to exploit a massively parallel computer with distri...

Data mining and probabilistic models for error estimate analysis of finite element method

In this paper, we propose a new approach based on data mining techniques and probabilistic models to compare and analyze finite element results of partial differential equations. We focus on the numerical errors produced by linear and quadratic finite element approximations. We first show how error estimates contain a kind of numerical uncertainty in their evaluation, which may influence and even damage the precision of finite element numerical results. A model problem, derived from an elliptic approximate Vlasov–Maxwell system, is then introduced. We define some variables as physical predictors, and we characterize how they influence the odds of the linear and quadratic finite elements to be locally “same order” accurate. Beyond this example, this approach proposes a method to compare, between several approximation methods, the accuracy of numerical results.

Residual-based a posteriori error estimate for interface problems: Nonconforming linear elements

In this paper, we study a modified residual-based a posteriori error estimator for the nonconforming linear finite element approximation to the interface problem. The reliability of the estimator is analyzed by a new and direct approach without using the Helmholtz decomposition. It is proved that the estimator is reliable with constant independent of the jump of diffusion coefficients across the interfaces, without the assumption that the diffusion coefficient is quasi-monotone. Numerical results for one test problem with intersecting interfaces are also presented.

Full Discretization of Semilinear Stochastic Wave Equations Driven by Multiplicative Noise

A fully discrete approximation of the semilinear stochastic wave equation driven by multiplicative noise is presented. A standard linear finite element approximation is used in space, and a stochastic trigonometric method is used for the temporal approximation. This explicit time integrator allows for mean-square error bounds independent of the space discretization and thus does not suffer from a step size restriction as in the often used Stormer-Verlet leapfrog scheme. Furthermore, it satisfies an almost trace formula (i.e., a linear drift of the expected value of the energy of the problem). Numerical experiments are presented and confirm the theoretical results.

Computable Error Estimates for Finite Element Approximations of Elliptic Partial Differential Equations with Rough Stochastic Data

We derive computable error estimates for finite element approximations of linear elliptic partial differential equations with rough stochastic coefficients. In this setting, the exact solutions contain high frequency content that standard a posteriori error estimates fail to capture. We propose goal-oriented estimates, based on local error indicators, for the pathwise Galerkin and expected quadrature errors committed in standard, continuous, piecewise linear finite element approximations. Derived using easily validated assumptions, these novel estimates can be computed at a relatively low cost and have applications to subsurface flow problems in geophysics where the conductivities are assumed to have lognormal distributions with low regularity. Our theory is supported by numerical experiments on test problems in one and two dimensions.