Mixed finite element methods for computing groundwater velocities

Central to the understanding of problems in water quality and quantity for effective management of water resources is the development of accurate numerical models to stimulate groundwater flows and contaminant transfer. We discuss several important difficulties arising in modeling of subsurface flow and present promising numerical procedures for alleviating these problems. Furthermore, we describe mixed-finite element techniques for accurately approximating fluid velocities, and review computational results on a variety of hydrologic problems.     

Adaptive finite element methods for Boussinesq equations

An a posteriori error analysis for Boussinesq equations is derived in this article. Then we compare this new estimate with a previous one developed for a regularized version of Boussinesq equations in a previous work. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 214–236, 2000     

A computer-assisted proof for photonic band gaps

We investigate photonic crystals, modeled by a spectral problem for Maxwell’s equations with periodic electric permittivity. Here, we specialize to a two-dimensional situation and to polarized waves. By Floquet–Bloch theory, the spectrum has band-gap structure, and the bands are characterized by families of eigenvalue problems on a periodicity cell, depending on a parameter k varying in the Brillouin zone K. We propose a computer-assisted method for proving the presence of band gaps: For k in a finite grid in K, we obtain eigenvalue enclosures by variational methods supported by finite element computations, and then capture all k ∈ K by a perturbation argument.     

A finite element frequency domain method for 2D photonic crystals

In this paper we propose a new finite element frequency domain (FEFD) method to compute the band spectra of 2D photonic crystals without impurities. Exploiting periodicity to identify discretization points differing by a period, it increases the effectiveness of the algorithm and reduces significantly its computational complexity. The results of an extensive experimentation indicate that our method offers an effective alternative to the most quoted methods in the literature.     

Adaptive space-time finite element methods for dynamic linear thermoelasticity

This article focuses on goal oriented error control for dynamic linear thermoelastic problems. To this end, we present a space-time formulation of this problem class. The corresponding space-time Galerkin discretization is the basis for the derivation of the error estimator using the dual weighted residual (DWR) method. A numerical example substantiates the accuracy and efficiency of the presented approach. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Adaptive finite element methods for Cahn–Hilliard equations

We develop a method for adaptive mesh refinement for steady state problems that arise in the numerical solution of Cahn–Hilliard equations with an obstacle free energy. The problem is discretized in time by the backward-Euler method and in space by linear finite elements. The adaptive mesh refinement is performed using residual based a posteriori estimates; the time step is adapted using a heuristic criterion. We describe the space–time adaptive algorithm and present numerical experiments in two and three space dimensions that demonstrate the usefulness of our approach.     

Adaptive finite element methods for elliptic equations with non-smooth coefficients

Summary. We consider a second-order elliptic equation with discontinuous or anisotropic coefficients in a bounded two- or three dimensional domain, and its finite-element discretization. The aim of this paper is to prove some a priori and a posteriori error estimates in an appropriate norm, which are independent of the variation of the coefficients. Received February 5, 1999 / Published online March 16, 2000     

Adaptive finite element methods for elliptic equations over hierarchical T-meshes

Isogeometric analysis using NURBS (Non-Uniform Rational B-Splines) as basis functions gives accurate representation of the geometry and the solution but it is not well suited for local refinement. In this paper, we use the polynomial splines over hierarchical T-meshes (PHT-splines) to construct basis functions which not only share the nice smoothness properties as the B-splines, but also allow us to effectively refine meshes locally. We develop a residual-based a posteriori error estimate for the finite element discretization of elliptic equations using PHT-splines basis functions and study their approximation properties. In addition, we conduct numerical experiments to verify the theory and to demonstrate the effectiveness of the error estimate and the high order approximations provided by the numerical solution.     

Adaptive finite element methods for elliptic equations over hierarchical T-meshes

Isogeometric analysis using NURBS (Non-Uniform Rational B-Splines) as basis functions gives accurate representation of the geometry and the solution but it is not well suited for local refinement. In this paper, we use the polynomial splines over hierarchical T-meshes (PHT-splines) to construct basis functions which not only share the nice smoothness properties as the B-splines, but also allow us to effectively refine meshes locally. We develop a residual-based a posteriori error estimate for the finite element discretization of elliptic equations using PHT-splines basis functions and study their approximation properties. In addition, we conduct numerical experiments to verify the theory and to demonstrate the effectiveness of the error estimate and the high order approximations provided by the numerical solution.     

Modified edge finite elements for photonic crystals

We consider Maxwell’s equations with periodic coefficients as it is usually done for the modeling of photonic crystals. Using Bloch/Floquet theory, the problem reduces in a standard way to a modification of the Maxwell cavity eigenproblem with periodic boundary conditions. Following [8], a modification of edge finite elements is considered for the approximation of the band gap. The method can be used with meshes of tetrahedrons or parallelepipeds. A rigorous analysis of convergence is presented, together with some preliminary numerical results in 2D, which fully confirm the robustness of the method. The analysis uses well established results on the discrete compactness for edge elements, together with new sharper interpolation estimates.     

Adaptive finite element methods for the identification of distributed parameters in elliptic equation

In this paper, adaptive finite element method is developed for the estimation of distributed parameter in elliptic equation. Both upper and lower error bound are derived and used to improve the accuracy by appropriate mesh refinement. An efficient preconditioned project gradient algorithm is employed to solve the nonlinear least-squares problem arising in the context of parameter identification problem. The efficiency of our error estimators is demonstrated by some numerical experiments.      

Adaptive finite element methods for conservation laws based on a posteriori error estimates

We prove a posteriori error estimates for a finite element method for systems of strictly hyperbolic conservation laws in one space dimension, and design corresponding adaptive methods. The proof of the a posteriori error estimates is based on a strong stability estimate for an associated dual problem, together with the Galerkin orthogonality of the finite-element method. The strong stability estimate uses the entropy condition for the system in an essential way. ©1995 John Wiley & Sons, Inc.     

Quasi-conforming element techniques for penalty finite element methods

The Quasi-Conforming Element (QCE) technique is introduced in this paper for calculating Penalty Finite Element problems. Unlike the Reduced Integration methods, the QCE technique uses multiple sets of functions to approximate strains and is independent of the integration order. This technique is applied to the incompressible linear elastic problem and the medium thickness plate bending problem by means of examples, and the numerical results are shown. The related generalized variational principle is given.     

Locking-free finite element methods for shells

We propose a new family of finite element methods for the Naghdi shell model, one method associated with each nonnegative integer . The methods are based on a nonstandard mixed formulation, and the th method employs triangular Lagrange finite elements of degree augmented by bubble functions of degree for both the displacement and rotation variables, and discontinuous piecewise polynomials of degree for the shear and membrane stresses. This method can be implemented in terms of the displacement and rotation variables alone, as the minimization of an altered energy functional over the space mentioned. The alteration consists of the introduction of a weighted local projection into part, but not all, of the shear and membrane energy terms of the usual Naghdi energy. The relative error in the method, measured in a norm which combines the norm of the displacement and rotation fields and an appropriate norm of the shear and membrane stress fields, converges to zero with order uniformly with respect to the shell thickness for smooth solutions, at least under the assumption that certain geometrical coefficients in the Nagdhi model are replaced by piecewise constants.

     

Adaptive finite element procedures for elastoplastic problems at finite strains

A major difficulty in the context of adaptive analysis of geometrically nonlinear problems is to provide a robust remeshing procedure that accounts both for the error caused by the spatial discretization and for the error due to the time discretization. For stability problems, such as strain localization and necking, it is essential to provide a step–size control in order to get a robust algorithm for the solution of the boundary value problem. For this purpose we developed an easy to implement step–size control algorithm. In addition we will consider possible a posteriori error indicators for the spatial error distribution of elastoplastic problems at finite strains. This indicator is adopted for a density–function–based adaptive remeshing procedure. Both error indicators are combined for the adaptive analysis in time and space. The performance of the proposed method is documented by means of representative numerical examples.     

Nonconforming finite element methods

Some interesting and important nonconforming finite elements for the second- and fourth-order elliptic problems are briefly described and analyzed.     

Stopping criteria for iterations in finite element methods

Summary. This work extends the results of Arioli [1], [2] on stopping criteria for iterative solution methods for linear finite element problems to the case of nonsymmetric positive-definite problems. We show that the residual measured in the norm induced by the symmetric part of the inverse of the system matrix is relevant to convergence in a finite element context. We then use Krylov solvers to provide alternative ways of calculating or estimating this quantity and present numerical experiments which validate our criteria.Mathematics Subject Classification (2000): 65N30, 65F10, 65F35     

Adaptive finite element methods for mixed control-state constrained optimal control problems for elliptic boundary value problems

Mixed control-state constraints are used as a relaxation of originally state constrained optimal control problems for partial differential equations to avoid the intrinsic difficulties arising from measure-valued multipliers in the case of pure state constraints. In particular, numerical solution techniques known from the pure control constrained case such as active set strategies and interior-point methods can be used in an appropriately modified way. However, the residual-type a posteriori error estimators developed for the pure control constrained case can not be applied directly. It is the essence of this paper to show that instead one has to resort to that type of estimators known from the pure state constrained case. Up to data oscillations and consistency error terms, they provide efficient and reliable estimates for the discretization errors in the state, a regularized adjoint state, and the control. A documentation of numerical results is given to illustrate the performance of the estimators.     

Interior estimates for nonconforming finite element methods

Summary. Interior error estimates are derived for a wide class of nonconforming finite element methods for second order scalar elliptic boundary value problems. It is shown that the error in an interior domain can be estimated by three terms: the first one measures the local approximability of the finite element space to the exact solution, the second one measures the degree of continuity of the finite element space (the consistency error), and the last one expresses the global effect through the error in an arbitrarily weak Sobolev norm over a slightly larger domain. As an application, interior superconvergences of some difference quotients of the finite element solution are obtained for the derivatives of the exact solution when the mesh satisfies some translation invariant condition. Received December 29, 1994