Piecewise Linear Finite Element Approximation Research Articles

A posteriori error estimates in finite element solution of structure vibration problems with applications to acoustical fluid-structure analysis

We present an a posteriori error estimator for piecewise linear finite element approximations of structure vibration problems. We prove that this estimator is equivalent to the energy norm of the error. Also, we introduce an adaptive mesh-refinement procedure based on the proposed estimator to analize fluid-structure interaction problems. Finally, numerical results for some test examples are presented which show the efficiency of the error estimator and the mesh-refinement techniques.

Two-level Schwarz method for unilateral variational inequalities

The numerical solution of variational inequalities of obstacle type associated with second-order elliptic operators is considered. Iterative methods based on the domain decomposition approach are proposed for discrete obstacle problems arising from the continuous, piecewise linear finite element approximation of the differential problem. A new variant of the Schwarz methodology, called the two-level Schwarz method, is developed offering the possibility of making use of fast linear solvers (e.g., linear multigrid and fictitious domain methods) for the genuinely nonlinear obstacle problems. Namely, by using particular monotonicity results, the computational domain can be partitioned into (mesh) subdomains with linear and nonlinear (obstacle-type) subproblems. By taking advantage of this domain decomposition and fast linear solvers, efficient implementation algorithms for large-scale discrete obstacle problems can be developed. The last part of the paper is devoted to illustrate numerical experiments.

Finite element approximation of the Cahn-Hilliard equation with concentration dependent mobility

We consider the Cahn-Hilliard equation with a logarithmic free energy and non-degenerate concentration dependent mobility. In particular we prove that there exists a unique solution for sufficiently smooth initial data. Further, we prove an error bound for a fully practical piecewise linear finite element approximation in one and two space dimensions. Finally some numerical experiments are presented.

Finite element approximation of a semilinear elliptic problem with a singular nonlinearity

Given $ \nu \in{\Bbb R}_{+} $ , we consider the following problem: find $u > 0$ , such that \begin{displaymath} -\Delta u = cu^{-\nu} \quad \mbox{in}\,\, \Omega\,, \quad u = 0 \quad \mbox{on}\,\, \partial \Omega\,, \end{displaymath} where $\Omega \subset{\Bbb R}^{d},\, d = 1,\,2$ or 3, and $c > 0$ in $\bar{\Omega}$ . We prove $H^{1}$ and $L^{\infty}$ error bounds for the standard continuous piecewise linear Galerkin finite element approximation with a (weakly) acute triangulation. Our bounds are nearly optimal. In addition, for d = 1 and 2 and $c \in{\Bbb R}_{+}$ we analyze a more practical scheme involving numerical integration on the nonlinear term. We obtain nearly optimal $H^{1}$ and $L^{\infty}$ error bounds for d = 1. For this case we also present some numerical results.

An improved error bound for a finite element approximation of a model for phase separation of a multi-component alloy

Using the approach of Rulla (1996 SIAM J. Numer. Anal. 33, 68-87) for analysing the time discretization error and assuming more regularity on the initial data, we improve on the error bound derived by Barrett and Blowey (1996 IMA J. Numer. Anal. 16, 257-287) for a fully practical piecewise linear finite element approximation with a backward Euler time discretization of a model for phase separation of a multi-component alloy.

Schwarz algorithm for the solution of variational inequalities with nonlinear source terms

Numerical solution of variational inequalities of obstacle type associated with some free boundary problems with nonlinear source terms is considered. Iterative methods based on the combination of the domain decomposition technique and the Newton-like approach are proposed for solving discrete obstacle problems arising from the continuous, piecewise linear finite element approximation of the problems. The calculation processes are easily to be parallelized. The convergence and specially the convergent rate of the algorithms are obtained.

Finite element approximation of a model for phase separation of a multi-component alloy with a concentration-dependent mobility matrix

We consider a model for phase separation of a multi-component alloy with a concentration-dependent mobility matrix and logarithmic free energy. In particular we prove that there exists a unique solution for sufficiently smooth initial data. Further, we prove an error bound for a fully practical piecewise linear finite element approximation in one and two space dimensions. Finally numerical experiments with three components in one space dimension are presented.

Superconvergent recovery of gradients of piecewise linear finite element approximations on non‐uniform mesh partitions

We propose the use of an averaging scheme, which recovers gradients from piecewise linear finite element approximations on the (1 + α˜)—regular triangular elements to gradients of the weak solution of a second-order elliptic boundary value problem in the 2-dimensional space. The recovered gradients, from mid-points of element edges, are superconvergent estimates of the true gradients. This work is an extension of Levine [Levine, IMA J. Numer. Anal. 5, 407 (1985)] and follows some of the ideas therein. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14:169–192, 1998

Superconvergent recovery of gradients of piecewise linear finite element approximations on non-uniform mesh partitions

We propose the use of an averaging scheme, which recovers gradients from piecewise linear finite element approximations on the (1 + α˜)—regular triangular elements to gradients of the weak solution of a second-order elliptic boundary value problem in the 2-dimensional space. The recovered gradients, from mid-points of element edges, are superconvergent estimates of the true gradients. This work is an extension of Levine [Levine, IMA J. Numer. Anal. 5, 407 (1985)] and follows some of the ideas therein. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14:169–192, 1998

Finite element approximation of a model for phase separation of a multi-component alloy with non-smooth free energy

A model for the phase separation of a multi-component alloy with non-smooth free energy is considered. An error bound is proved for a fully practical piecewise linear finite element approximation using a backward Euler time discretization. An iterative scheme for solving the resulting nonlinear algebraic system is analysed. Finally numerical experiments with three components in one and two space dimensions are presented. In the one dimensional case we compare some steady states obtained numerically with the corresponding stationary solutions of the continuous problem, which we construct explicitly.

An error bound for the finite element approximation of a model for phase separation of a multi-component alloy

An error bound is proved for a fully practical piecewise linear finite element approximation, using a backward-Euler time discretization, of a model for phase separation of a multi-component alloy. Numerical experiments with three components in one and two space dimensions are also presented.

Finite Element Approximation of Some Degenerate Monotone Quasilinear Elliptic Systems

In this paper we examine the continuous piecewise linear finite element approximation of the following system: given ${\bf f} \equiv (f_j )$ and ${\bf g} \equiv (g_j )$, find ${\bf u} \equiv (u_j )$($(j = 1 \to r$ with $r = 1$ or 2) such that $ - \nabla \cdot (K(z,\nabla {\bf u}(z))\nabla {\bf u}(z)) = {\bf f}(z),\quad z \in \Omega \subset R^2 ,\quad {\bf u} |_{\partial \Omega } = {\bf g} |_{\partial \Omega } ,$ where $(\nabla {\bf u}) \equiv {{\partial u_j } / {\partial z_i }}\, 1 \leq i \leq 2,\, 1 \leq j \leq r$ and K is a given matrix on $\Omega \times R^{2 \times r} $. We characterize a class of matrices K for which we prove error bounds for this discretization. For sufficiently regular solutions ${\bf u}$, achievable at least for some model problems, our bounds improve on existing results in the literature. It is shown that for a notable subclass of K, for which only suboptimal error bounds have been previously derived, the piecewise linear finite element approximation of this problem will converge at the optimal rate in an energy-type norm. It is also shown that the techniques used in this paper can be applied to more general problems.

An error bound for the finite element approximation of the Cahn-Hilliard equation with logarithmic free energy

An error bound is proved for a fully practical piecewise linear finite element approximation, using a backward Euler time discretization, of the Cahn-Hilliard equation with a logarithmic free energy.

The finite element approximation of a coupled reaction-diffusion problem with non-Lipschitz nonlinearities

A coupled semilinear elliptic problem modelling an irreversible, isothermal chemical reaction is introduced, and discretised using the usual piecewise linear Galerkin finite element approximation. An interesting feature of the problem is that a reaction order of less than one gives rise to a "dead core" region. Initially, one reactant is assumed to be acting as a catalyst and is kept constant. It is shown that error bounds previously obtained for a scheme involving numerical integration can be improved upon by considering a quadratic regularisation of the nonlinear term. This technique is then applied to the full coupled problem, and optimal $H^1$ and $L^\infty$ error bounds are proved in the absence of quadrature. For a scheme involving numerical integration, bounds similar to those obtained for the catalyst problem are shown to hold.

Quasi-norm error bounds for the finite element approximation of some degenerate quasilinear parabolic equations and variational inequalities

In this paper we study the continuous piecewise linear finite element approximation of the following problem: Let Ω be an open set in Rd with d=l or 2. Given T>0, f and uO; find u∊K, where K is a closed convex subset of the So bo lev space , such that for x∊Ω and for any v∊K for a.e. t∊(0,T], where k∊C(0,∞) is a given nonnegative function with k(s)s strictly increasing for s≥O, but possibly degenerate, and p∊(1,∞) depends on k. For such a general problem we establish error bounds in energy type norms for a fully discrete approximation based on the backward Euler time discretisation. We show that these error bounds converge at the optimal rate with respect to the space discretisation, provided p≤2 and the solution u is sufficiently regular.

Finite element approximation of a model vortex problem

The purpose of this paper is to prove error bounds for the standard Galerkin piecewise linear finite element approximation of the following Model Vortex Problem: Given a bounded, connected domain with boundary ∂Ω e C 2,$gamma; for some γ > 0, and constants p e [0,1) and find u such that The main mathematical interest of P lies in the fact that it is a semilinear elliptic free boundary problem with a non–Lipschitz nonlinearity. The existence of a well-defined maximal branch of solutions of P is shown for d sufficiently small. Using a monotonicity and maximum principle approach, an L ∞ error bound of order , where, δ e (0,1), is proved for a finite element approximation of P involving exact integration of the nonlinear term, provided that the maximal solution branch of the continuous problem is Holder continuous with exponent with respect to d. This condition is then shown to hold for any δ e (0,1), for p e (0,1) for and d sufficiently small. Thus the stated error bound is near-optimal.

A mixed problem for electrostatic potential in semiconductors

AbstractIn this article we deal with the solution in Ω ⊂ R2 of the quasi linear equation −Δu = f(x, y, u(x, y)) subject to mixed boundary data and representing Gauss' law in a semiconductor device, where u and f are, respectively, the electrostatic potential and the space charge density after a suitable scaling. In the following we consider the associated variational problem of finding in a suitable subspace of H1(Ω) the minimum of the functional \documentclass{article}\pagestyle{empty}\begin{document}$ J(u)\, = \,\int {_\Omega } (\frac{1}{2}\left| {\nabla u\left| {^2 \, - \,{\cal F}(x,y,u)\,d\Omega,} \right.} \right. $\end{document}, where \documentclass{article}\pagestyle{empty}\begin{document}$ {\cal F}(x,y,u)\, = \,\int f (x,y,\xi)\,d\xi, $\end{document} and we prove existence and uniqueness of a weak solution according to the technique of Convex Analysis. The numerical study is then carried on by a piecewise linear finite element approximation, which is proved to converge in the H1‐norm to the exact solution of the variational problem; some numerical examples are also included. © 1994 John Wiley & Sons, Inc.

Quasi-norm error bounds for the finite element approximation of some degenerate quasilinear elliptic equations and variational inequalities

In this paper we study the continuous piecewise linear finite element approximation of the following problem: Let Ω be an open set in Rd with d=l or 2. Given T>0, f and uO; find u∊K, where K is a closed convex subset of the So bo lev space , such that for x∊Ω and for any v∊K for a.e. t∊(0,T], where k∊C(0,∞) is a given nonnegative function with k(s)s strictly increasing for s≥O, but possibly degenerate, and p∊(1,∞) depends on k. For such a general problem we establish error bounds in energy type norms for a fully discrete approximation based on the backward Euler time discretisation. We show that these error bounds converge at the optimal rate with respect to the space discretisation, provided p≤2 and the solution u is sufficiently regular.

A Fully Discrete Adaptive Nonlinear Chernoff Formula

An adaptive piecewise linear finite element approximation to a linear method, the so-called nonlinear Chernoif formula, for the simplest two-phase Stefan problem in two dimensions are discussed. The full discretization amounts to solving linear positive definite symmetric systems, followed by an explicit nodewise algebraic correction. Consecutive meshes are highly graded and noncompatible. The pointwise information needed to generate a new mesh is extracted from both the discrete enthalpy $U^n $ and temperature $\Theta ^n $. For well-behaved discrete transition layers $\mathcal{T}^n = \{ 0 < U^n - \Theta ^n < 1\} $, triangles are expected to be $\mathcal{O}(\tau )$ within a thin strip of width $\mathcal{O}(\tau ^{{1 / 2}} )$, the so-called refined region, which in turn must contain the transition layer. The local meshsize increases up to $\mathcal {O}(\tau ^{1/2} )$ elsewhere; here $\tau $ stands for the uniform time-step. The expected number of spatial degrees of freedom becomes $\mathcal {O}(\tau ^{ - 3/2} )$, which compares quite favorably with that required without adaptivity, namely $\mathcal {O}(\tau ^{ - 2} )$. Stability is examined in a number of Sobolev norms and then is used to derive a quasi-optimal rate of convergence of order essentially $\mathcal {O}(\tau ^{1/2} )$ in the natural energy spaces. Numerical experiments illustrate the performance of the proposed method.

Error bounds for the finite element approximation of a degenerate quasilinear parabolic variational inequality

In this paper, we establish some error bounds for the continuous piecewise linear finite element approximation of the following problem: Let Ω be an open set in ℝd, withd=1 or 2. GivenT>0,p ∈ (1, ∞),f andu0; findu ∈K, whereK is a closed convex subset of the Sobolev spaceW01,p(Ω), such that for anyv∈K $$\begin{gathered} \int\limits_\Omega {u_1 (\upsilon - u) dx + } \int\limits_\Omega {\left| {\nabla u} \right|^{p - 2} } \nabla u \cdot \nabla (\upsilon - u) dx \geqslant \int\limits_\Omega {f(\upsilon - u) dx for} a.e. t \in (0,T], \hfill \\ u = 0 on \partial \Omega \times (0,T] and u(0,x) = u_0 (x) for x \in \Omega . \hfill \\ \end{gathered} $$ We prove error bounds in energy type norms for the fully discrete approximation using the backward Euler time discretisation. In some notable cases, these error bounds converge at the optimal rate with respect to the space discretisation, provided the solutionu is sufficiently regular.