Linear Elliptic Problems Research Articles

A Least-Squares/Fictitious Domain Method for Linear Elliptic Problems with Robin Boundary Conditions

AbstractIn this article, we discuss a least-squares/fictitious domain method for the solution of linear elliptic boundary value problems with Robin boundary conditions. Let Ω and ω be two bounded domains of Rd such that ω̅⊂Ω. For a linear elliptic problem in Ω\ω̅ with Robin boundary condition on the boundary ϒ of ω, our goal here is to develop a fictitious domain method where one solves a variant of the original problem on the full Ω, followed by a well-chosen correction over ω. This method is of the virtual control type and relies on a least-squares formulation making the problem solvable by a conjugate gradient algorithm operating in a well chosen control space. Numerical results obtained when applying our method to the solution of two-dimensional elliptic and parabolic problems are given; they suggest optimal order of convergence.

An efficient, reliable and robust error estimator for elliptic problems in [formula omitted

In this article, we develop and analyze a hierarchical-type error estimator for a general class of second-order linear elliptic boundary value problems in bounded three-dimensional domains. This type of indicator automatically satisfies a global lower bound inequality, thereby giving efficiency, without regularity assumptions beyond those giving well-posedness of the continuous and discrete problems. The main focus of the paper is then to establish the reverse reliability result: a global upper bound on the error in terms of the error estimate (plus an oscillation term), again without additional regularity assumptions. The proof of this inequality depends on a clever choice of the space in which the error indicator lies and a moment-capturing quasi-interpolation result. We finish the article with a series of numerical experiments to illustrate the behavior predicted by the theoretical results.

Asymptotic approximations for solutions to quasilinear and linear elliptic problems with different perturbed boundary conditions in perforated domains

Asymptotic approximations for solutions to quasilinear and linear elliptic problems with different perturbed boundary conditions in perforated domains

On linear and nonlinear elliptic boundary value problems in the plane with discontinuous coefficients

Global Holder regularity of the gradient in Morrey spaces is established for planar elliptic discontinuous equations, estimating in an explicit way the Holder exponent in terms of the eigenvalues of the operator. The result is proved for Dirichlet or normal derivative problems and for nonlinear operators.

A Finite Element Method for Second Order Nonvariational Elliptic Problems

We propose a numerical method to approximate the solution of second order elliptic problems in nonvariational form. The method is of Galerkin type using conforming finite elements and applied directly to the nonvariational (nondivergence) form of a second order linear elliptic problem. The key tools are an appropriate concept of a 'finite element Hessian' and a Schur complement approach to solving the resulting linear algebra problem. The method is illustrated with computational experiments on three linear and one quasilinear PDE, all in nonvariational form.

Composite Finite Elements for Elliptic Interface Problems

AbstractWe present a Composite Finite Element Method for the approximation of linear elliptic boundary value problems of Dirichlet type with discontinuous coefficients. The challenge is the discontinuity of the coefficient (interface) which is not necessarily resolved by the underlying finite element mesh. The method is non‐conforming in the sense that shape functions preserve continuity across the interface only in an approximative way. However, the construction allows to balance the non‐conformity and the overall discretization error. (© 2010 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Convergence and quasi-optimality of adaptive FEM for Steklov eigenvalue problems

The main goal of this article is the study of the convergence and optimality properties of adaptive finiteelement methods (AFEMs) for Steklov eigenvalue problems. These problems arise, for example, in the analysis of the stability of mechanical oscillators immersed in a viscous fluid (see Conca et al., 1995 and the references therein) or in the study of vibration modes of a structure interacting with an incompressible fluid (Bermudez et al., 2000). Finite-element approximations for these problems have been widely used and analysed under a general framework. AFEMs make efficient use of the computational resources, and for certain problems AFEMs are even indispensable for their numerical resolvability. The ultimate goal of adaptive methods is to equidistribute the error and the computational effort to obtain a sequence of meshes with optimal complexity. Historically, the first step to prove optimality has been to understand the convergence of adaptive methods. A basic result for the convergence of linear problems was presented in Morin et al. (2008), where very general conditions on the linear problems and on the ingredients of the AFEM that guarantee convergence were stated. Following these ideas, the (plain) convergence of an AFEM for elliptic eigenvalue problems was proved in Garau et al. (2009). The optimality of an AFEM using Dorfler’s marking strategy (Dorfler, 1996) was proved by Stevenson (2007) and Cascon et al. (2008) for linear elliptic problems. The linear convergence of an AFEM for elliptic eigenvalue problems was proved in Giani & Graham (2009) when considering the approximation of simple eigenvalues, provided that the initial mesh is sufficiently fine, and an improvement of this AFEM including optimality was presented in Dai et al. (2008). In this article we consider the Steklov eigenvalue problem that consists of finding λ ∈ R and u 6≡ 0 such that −∇ ∙ (A∇u)+ cu = 0 in Ω, (A∇u) ∙ −→n = λρu on ∂Ω,

Multiplicity Results for a Class of Asymptotically Linear Elliptic Problems With Resonance and Applications to Problems With Measure Data

Abstract We study existence and multiplicity results for solutions of elliptic problems of the type -Δu = g(x; u) in a bounded domain Ω with Dirichlet boundary conditions. The function g(x; s) is asymptotically linear as |s| → +∞. Also resonant situations are allowed. We also prove some perturbation results for Dirichlet problems of the type -Δu = gƐ(x, u) where gε(x, s) → g(x, s) as ε → 0. The previous results find an application in the study of Dirichlet problems of the type -Δu = g(x, u)+μ where μ is a Radon measure. To properly set the above mentioned problems in a variational framework we also study existence and properties of critical points of a class of abstract nonsmooth functional defined on Banach spaces and extend to this nonsmooth framework some classical linking theorems.

On variational eigenvalues of the p -Laplacian which are not of Ljusternik-Schnirelmann type

In this paper we demonstrate the fact that the famous Ljusternik–Schnirelmann characterization of some eigenvalues of nonlinear elliptic problems (by a minimax formula) has a global variational character. Indeed, we show that, for some homogeneous quasilinear elliptic eigenvalue problems, there are variational eigenvalues other than those of the Ljusternik–Schnirelmann type. In contrast, these eigenvalues have a local variational character. Such a phenomenon does not occur in typical linear elliptic eigenvalue problems, owing to the Courant–Fischer theorem which is the linear analogue and predecessor of the Ljusternik–Schnirelmann theory.

Numerical Analysis of Nonlinear Eigenvalue Problems

We provide a priori error estimates for variational approximations of the ground state energy, eigenvalue and eigenvector of nonlinear elliptic eigenvalue problems of the form −div(A ∇ u)+Vu+f(u 2)u=λ u, $\|u\|_{L^{2}}=1$ . We focus in particular on the Fourier spectral approximation (for periodic problems) and on the ℙ1 and ℙ2 finite-element discretizations. Denoting by (u δ ,λ δ ) a variational approximation of the ground state eigenpair (u,λ), we are interested in the convergence rates of $\|u_{\delta}-u\|_{H^{1}}$ , $\|u_{\delta}-u\|_{L^{2}}$ , |λ δ −λ|, and the ground state energy, when the discretization parameter δ goes to zero. We prove in particular that if A, V and f satisfy certain conditions, |λ δ −λ| goes to zero as $\|u_{\delta}-u\|_{H^{1}}^{2}+\|u_{\delta}-u\|_{L^{2}}$ . We also show that under more restrictive assumptions on A, V and f, |λ δ −λ| converges to zero as $\|u_{\delta}-u\|_{H^{1}}^{2}$ , thus recovering a standard result for linear elliptic eigenvalue problems. For the latter analysis, we make use of estimates of the error u δ −u in negative Sobolev norms.

Weak optimal controls in coefficients for linear elliptic problems

In this paper we study an optimal control problem associated to a linear degenerate elliptic equation with mixed boundary conditions. The equations of this type can exhibit the Lavrentieff phenomenon and non-uniqueness of weak solutions. We adopt the weight function as a control in L 1(Ω). Using the direct method in the Calculus of variations, we discuss the solvability of this optimal control problem in the class of weak admissible solutions.

ℎ𝑝-Optimal discontinuous Galerkin methods for linear elliptic problems

The aim of this paper is to present and analyze a class ofhphp-version discontinuous Galerkin (DG) discretizations for the numerical approximation of linear elliptic problems. This class includes a number of well-known DG formulations. We will show that the methods are stable provided that the stability parameters are suitably chosen. Furthermore, on (possibly irregular) quadrilateral meshes, we shall prove that the schemes converge all optimally in the energy norm with respect to both the local element sizes and polynomial degrees provided that homogeneous boundary conditions are considered.

Fully Computable Error Bounds for Discontinuous Galerkin Finite Element Approximations on Meshes with an Arbitrary Number of Levels of Hanging Nodes

We obtain fully computable a posteriori error bounds on the broken energy seminorm and discontinuous Galerkin norm (DG-norm) of the error in first order symmetric interior penalty Galerkin (SIPG), nonsymmetric interior penalty Galerkin (NIPG), and incomplete interior penalty Galerkin (IIPG) finite element approximations of a linear second order elliptic problem on meshes containing an arbitrary number of levels of hanging nodes and comprised of triangular elements. The estimators are completely free of unknown constants and provide guaranteed numerical bounds on the broken energy seminorm and DG-norm of the error. These estimators are also shown to provide a lower bound for the broken energy seminorm and DG-norm of the error up to a constant and higher order data oscillation terms. We also obtain an explicit computable bound for the value of the interior penalty parameter needed to ensure the existence of the discontinuous Galerkin finite element approximation for all versions of the method.

On the Iterative Solution Methods for Finite-Dimensional Inclusions with Applications to Optimal Control Problems

AbstractIterative methods for finite-dimensional inclusions which arise in applying a finite-element or a finite-difference method to approximate state-constrained optimal control problems have been investigated. Specifically, problems of control on the right- hand side of linear elliptic boundary value problems and observation in the entire domain have been considered. The convergence and the rate of convergence for the iterative algorithms based on the finding of the control function or Lagrange multipliers are proved.

A Priori Error Analysis for Linear Quadratic Elliptic Neumann Boundary Control Problems with Control and State Constraints

In this paper we consider a state-constrained optimal control problem with boundary control, where the state constraints are imposed only in an interior subdomain. Our goal is to derive a priori error estimates for a finite element discretization with and without additional regularization. We will show that the separation of the set where the control acts and the set where the state constraints are given improves the approximation rates significantly. The theoretical results are illustrated by numerical computations.

Flujos Isotérmicos/Térmicos, Incompresibles y Viscosos con Formulación Velocidad-Vorticidad

Numerical results in two dimensions for isothermal/thermal flows of viscous incompressible fluids mathematically modeled by the Navier-Stokes equations and the Bousssinesq approximation are presented in the velocity-vorticity formulation. The thermal flows correspond to natural and mixed convection. The results are obtained based on a fixed point iterative process for solving the nonlinear elliptic system that results after an appropriate time discretization is made. The iterative process leads to the solution of uncoupled, well conditioned linear elliptic problems for which very efficient solvers are known to exist regardless of the space discretization.

A new error analysis for discontinuous finite element methods for linear elliptic problems

The standard a priori error analysis of discontinuous Galerkin methods requires additional regularity on the solution of the elliptic boundary value problem in order to justify the Galerkin orthogonality and to handle the normal derivative on element interfaces that appear in the discrete energy norm. In this paper, a new error analysis of discontinuous Galerkin methods is developed using only the H k H^k weak formulation of a boundary value problem of order 2 k 2k . This is accomplished by replacing the Galerkin orthogonality with estimates borrowed from a posteriori error analysis and by using a discrete energy norm that is well defined for functions in H k H^k .

A polynomial chaos approach to stochastic variational inequalities

We consider stochastic elliptic variational inequalities of the second kind involving a bilinear form with stochastic diffusion coefficient. We prove existence and uniqueness of weak solutions, propose a stochastic Galerkin approximation of an equivalent parametric reformulation, and show equivalence to a related collocation method. Numerical experiments illustrate the efficiency of our approach and suggest similar error estimates as for linear elliptic problems.

On sign-changing solution for a fourth-order asymptotically linear elliptic problem

In this paper we deal with a fourth-order elliptic problem whose nonlinear term is asymptotically linear at both zero and infinity. By using the variational method, we obtain an existence result of sign-changing solutions as well as positive and negative solutions.

Constant free error bounds for nonuniform order discontinuous Galerkin finite-element approximation on locally refined meshes with hanging nodes

We obtain fully computable constant free a posteriori error bounds on the broken energy seminorm and the discontinuous Galerkin (DG) norm of the error for nonuniform polynomial order symmetric interior penalty Galerkin, nonsymmetric interior penalty Galerkin and incomplete interior penalty Galerkin finite-element approximations of a linear second-order elliptic problem on meshes containing hanging nodes and comprised of triangular elements. The estimators are completely free of unknown constants and provide guaranteed numerical bounds on the broken energy seminorm and the DG norm of the error. These estimators are also shown to provide a lower bound for the broken energy seminorm and the DG norm of the error up to a constant and higher-order data oscillation terms.