Two-Grid Discontinuous Galerkin Method for Quasi-Linear Elliptic Problems

In this paper, we consider the symmetric interior penalty discontinuous Galerkin (SIPG) method with piecewise polynomials of degree r≥1 for a class of quasi-linear elliptic problems in Ω⊂ℝ². We propose a two-grid approximation for the SIPG method which can be thought of as a type of linearization of the nonlinear system using a solution from a coarse finite element space. With this technique, solving a quasi-linear elliptic problem on the fine finite element space is reduced into solving a linear problem on the fine finite element space and solving the quasi-linear elliptic problem on a coarse space. Convergence estimates in a broken H ¹-norm are derived to justify the efficiency of the proposed two-grid algorithm. Numerical experiments are provided to confirm our theoretical findings. As a byproduct of the technique used in the analysis, we derive the optimal pointwise error estimates of the SIPG method for the quasi-linear elliptic problems in ℝ ^d ,d=2,3 and use it to establish the convergence of the two-grid method for problems in Ω⊂ℝ³.     

A variational approximation method for 2nd order elliptic eigenvalue problems in a composite structure with nonstandard boundary conditions

In this paper we deal with the finite element analysis of a class of eigenvalue problems (EVPs) in a composite structure in the plane, consisting of rectangular subdomains which enclose an intermediate region. Nonlocal boundary conditions (BCs) of Robin type are imposed on the inner boundaries, i.e. on the interfaces of the respective subdomains with the intermediate region. On the eventual interfaces between two subdomains we impose discontinuous transition conditions (TCs). Finally, we have classical local BCs at the outer boundaries. Such problems are related to some heat transfer problems e.g. in a horizontal cross section of a wall enclosing an air cave.     

A finite element approach to a quasi-variational inequality

We analyze the convergence of the finite element approximation to an elliptic one-dimensional quasi-variational inequality, connected to stochastic impulse control theory. We prove an optimal 0(h) error bound for the linear element solution of the associated variational selection. Then, by means of a continuity result, we derive anL ^∞-error estimate for the linear element solution of the quasi-variational inequality. Work supported by the Gruppo Nazionale per l'Analisi Matematica del C.N.R.     

Stopping criteria for iterative methods:¶applications to PDE's

We show that, when solving a linear system with an iterative method, it is necessary to measure the error in the space in which the residual lies. We present examples of linear systems which emanate from the finite element discretization of elliptic partial differential equations, and we show that, when we measure the residual in H ⁻¹(Ω), we obtain a true evaluation of the error in the solution, whereas the measure of the same residual with an algebraic norm can give misleading information about the convergence. We also state a theorem of functional compatibility that proves the existence of perturbations such that the approximate solution of a PDE is the exact solution of the same PDE perturbed. Received: March 2000 / Accepted: October 2000     

Looking for the best constant in a Sobolev inequality: a numerical approach

A numerical method for the computation of the best constant in a Sobolev inequality involving the spaces H ²(Ω) and C⁰([`(W)])C^{0}(\overline{\Omega}) is presented. Green’s functions corresponding to the solution of Poisson problems are used to express the solution. This (kind of) non-smooth eigenvalue problem is then formulated as a constrained optimization problem and solved with two different strategies: an augmented Lagrangian method, together with finite element approximations, and a Green’s functions based approach. Numerical experiments show the ability of the methods in computing this best constant for various two-dimensional domains, and the remarkable convergence properties of the augmented Lagrangian based iterative method.     

Résolution approchée de problèmes aux limites elliptiques par des schémas aux éléments finis a plusieurs fonctions arbitraires

Summary The present paper is devoted to the approximate solution of variational elliptic boundary value problems of the form: α(u, v)=(f, v)v ∈V by using approximations of the Hilbert spaceV with several degrees of freedom as constructed in a preceding paper [7]. These approximations lead to finite difference schemes involving several arbitrary parameters, whose solution converge to the exact solution of the boundary value problem if the values of these parameters are small enough. This fact can be utilized to diminish the error between the exact and the approximate solution by a suitable choice of these arbitrary parameters, so as to avoid the use of very small step lengths. The method may prove useful in cases where the coercivity constant of the bilinear form α (u, v) is small when compated to its continuity constant, and more generally for problems of the form: α (u, v)−λ (u. v.)=(f, v) where the constant λ is close to an eigenvalue of the boundary value problem.      

A posteriori error estimates for a mixed-FEM formulation of a non-linear elliptic problem

We consider the numerical solution, via the mixed finite element method, of a non-linear elliptic partial differential equation in divergence form with Dirichlet boundary conditions. Besides the temperature u and the flux $\text{σ}$, we introduce ∇u as a further unknown, which yields a variational formulation with a twofold saddle point structure. We derive a reliable a posteriori error estimate that depends on the solution of a local linear boundary value problem, which does not need any equilibrium property for its solvability. In addition, for specific finite element subspaces of Raviart–Thomas type we are able to provide a fully explicit a posteriori error estimate that does not require the solution of the local problems. Our approach does not need the exact finite element solution, but any reasonable approximation of it, such as, for instance, the one obtained with a fully discrete Galerkin scheme. In particular, we suggest a scheme that uses quadrature formulas to evaluate all the linear and semi-linear forms involved. Finally, several numerical results illustrate the suitability of the explicit error estimator for the adaptive computation of the corresponding discrete solutions.     

Numerical Blow-up of Semilinear Parabolic PDEs on Unbounded Domains in ℝ<Superscript>2</Superscript>

We study the numerical solution of semilinear parabolic PDEs on unbounded spatial domains Ω in ℝ² whose solutions blow up in finite time. Of particular interest are the cases where Ω=ℝ² or Ω is a sectorial domain in ℝ². We derive the nonlinear absorbing boundary conditions for corresponding, suitably chosen computational domains and then employ a simple adaptive time-stepping scheme to compute the solution of the resulting system of semilinear ODEs. The theoretical results are illustrated by a broad range of numerical examples.     

A stable nonconforming quadrilateral finite element method for the stationary Stokes and Navier–Stokes equations

Recently, Douglas et al. [4] introduced a new, low-order, nonconforming rectangular element for scalar elliptic equations. Here, we apply this element in the approximation of each component of the velocity in the stationary Stokes and Navier–Stokes equations, along with a piecewise-constant element for the pressure. We obtain a stable element in both cases for which optimal error estimates for the approximation of both the velocity and pressure in L ² can be established, as well as one in a broken H ¹-norm for the velocity. Received: January 1999 / Accepted: April 1999     

L ∞-error estimates of higher order mixed finite element approximations for elliptic optimal control problems

In this paper, we investigate the L ^∞-error estimates of the numerical solutions of linear-quadratic elliptic control problems by using higher order mixed finite element methods. The state and co-state are approximated by the order k Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise polynomials of order k (k≥1). Optimal L ^∞-error estimates are derived for both the control and the state approximations. These results are seemed to be new in the literature of the mixed finite element methods for optimal control problems.     

Superconvergence of mixed finite element methods on rectangular domains

Superconvergence by one power of h along Gauss lines for theBDFM (Brezzi-Douglas-Fortin-Marini) mixed finite element approximation by rectangular elements of the vector field associated with a second order elliptic equation on a rectangular domain is established under an appropriate assumption of smoothness on the exact solution.     

Newton Iterative Parallel Finite Element Algorithm for the Steady Navier-Stokes Equations

A combination method of the Newton iteration and parallel finite element algorithm is applied for solving the steady Navier-Stokes equations under the strong uniqueness condition. This algorithm is motivated by applying the Newton iterations of m times for a nonlinear problem on a coarse grid in domain Ω and computing a linear problem on a fine grid in some subdomains Ω _j ⊂Ω with j=1,…,M in a parallel environment. Then, the error estimation of the Newton iterative parallel finite element solution to the solution of the steady Navier-Stokes equations is analyzed for the large m and small H and h≪H. Finally, some numerical tests are made to demonstrate the the effectiveness of this algorithm.     

Numerical verification of solutions for elasto-plastic torsion problems

In this paper, we consider a numerical technique which enables us to verify the existence of solutions for the elasto-plastic torsion problems governed by the variational inequality. Based upon the finite element approximations and the explicit a priori error estimates for a simple problem, we present an effective verification procedure that through numerical computation generates a set which includes the exact solution. This paper is an extension of the previous paper [1] in which we mainly dealt with the obstacle problems, but some special techniques are utilized to verify the solutions for nondifferentiable nonlinear equations concerned with the present problem. A numerical example is illustrated.     

A Weak Galerkin Finite Element Method for a Type of Fourth Order Problem Arising from Fluorescence Tomography

In this paper, an innovative and effective numerical algorithm by the use of weak Galerkin (WG) finite element methods is proposed for a type of fourth order problem arising from fluorescence tomography. Fluorescence tomography is emerging as an in vivo non-invasive 3D imaging technique reconstructing images that characterize the distribution of molecules tagged by fluorophores. Weak second order elliptic operator and its discrete version are introduced for a class of discontinuous functions defined on a finite element partition of the domain consisting of general polygons or polyhedra. An error estimate of optimal order is derived in an \(H^2_{\kappa }\)-equivalent norm for the WG finite element solutions. Error estimates of optimal order except the lowest order finite element in the usual \(L^2\) norm are established for the WG finite element approximations. Numerical tests are presented to demonstrate the accuracy and efficiency of the theory established for the WG numerical algorithm.     

Geometry representation issues associated with p-version finite element computations

This paper addresses issues related to accurate geometry representation for p-version finite elements on curved three-dimensional domains. Specific options to account for domain geometry information during element-level computation are identified. Accuracy requirements on the geometry related approximations to preserve the optimal rate of finite element error convergence for second-order elliptic boundary value problems are given. An element geometric mapping scheme based on blending the exact shape of the domain boundary is described that can either be used directly during element integrations, or used to construct element-level geometric approximations of required accuracy. Smoothness issues of the rational blends on simplex topologies are discussed and a numerical example based on the solution of Poisson's equation in three dimensions is presented to illustrate the impact of the rational blends on the optimal rate of finite element error convergence.     

The Local Artificial Boundary Conditions for Numerical Simulations of the Flow Around a Submerged Body

We consider in this work the numerical approximations of the two-dimensional steady potential flow around a body moving in a liquid of finite constant depth at constant speed and distance below a free surface. Several vertical segments are introduced as the upstream and the downstream artificial boundaries, where a sequence of high-order local artificial boundary conditions are proposed. Then the original problem is solved in a finite computational domain, which is equivalent to a variational problem. The numerical approximations for the original problem are obtained by solving the variational problem with the finite element method. The numerical examples show that the artificial boundary conditions given in this work are very effective.     

A Finite Element Recovery Approach to Eigenvalue Approximations with Applications to Electronic Structure Calculations

In this paper, we propose and analyze a recovery approach for trilinear finite element approximations on locally-refined hexahedral meshes for a class of elliptic eigenvalue problems. In the approach a local high-order interpolation recovery is followed by some gradient averaging based defect correction scheme. It is proved theoretically and shown numerically that our recovery approach can produce highly accurate eigenpair approximations. And we observe from our numerical experiments that the recovered eigenvalue approximation from the gradient averaging based defect correction approximates the exact eigenvalue from below. Furthermore, this approach has been applied to electronic structure calculations to improve the total energy approximations with small extra overheads.     

The Streamline–Diffusion Method for Conforming and Nonconforming Finite Elements of Lowest Order Applied to Convection–Diffusion Problems

We consider the streamline-diffusion finite element method with finite elements of lowest order for solving convection-diffusion problems. Our investigations cover both conforming and nonconforming finite element approximations on triangular and quadrilateral meshes. Although the considered finite elements are of the same interpolation order their stability and approximation properties are quite different. We give a detailed overview on the stability and the convergence properties in the L ²- and in the streamline–diffusion norm. Numerical experiments show that often the theoretical predictions on the convergence properties are sharp. Received December 7, 1999; revised October 5, 2000     

Three-points interfacial quadrature for geometrical source terms on nonuniform grids

This paper deals with numerical (finite volume) approximations, on nonuniform meshes, for ordinary differential equations with parameter-dependent fields. Appropriate discretizations are constructed over the space of parameters, in order to guarantee the consistency in presence of variable cells’ size, for which L ^p -error estimates, 1≤p<+∞, are proven.     

Analysis of Galerkin Methods for the Fully Nonlinear Monge-Ampère Equation

This paper develops and analyzes finite element Galerkin and spectral Galerkin methods for approximating viscosity solutions of the fully nonlinear Monge-Ampère equation det (D ² u ⁰)=f (>0) based on the vanishing moment method which was developed by the authors in Feng and Neilan (J. Sci. Comput. 38:74–98, 2009) and Feng (Convergence of the vanishing moment method for the Monge-Ampère equation, submitted). In this approach, the Monge-Ampère equation is approximated by the fourth order quasilinear equation −εΔ² u ^ε +det D ² u ^ε =f accompanied by appropriate boundary conditions. This new approach enables us to construct convergent Galerkin numerical methods for the fully nonlinear Monge-Ampère equation (and other fully nonlinear second order partial differential equations), a task which has been impracticable before. In this paper, we first develop some finite element and spectral Galerkin methods for approximating the solution u ^ε of the regularized problem. We then derive optimal order error estimates for the proposed numerical methods. In particular, we track explicitly the dependence of the error bounds on the parameter ε, for the error u^e-u^e_hu^{\varepsilon}-u^{\varepsilon}_{h}. Due to the strong nonlinearity of the underlying equation, the standard error estimate technique, which has been widely used for error analysis of finite element approximations of nonlinear problems, does not work here. To overcome the difficulty, we employ a fixed point technique which strongly makes use of the stability property of the linearized problem and its finite element approximations. Finally, using the Argyris finite element method as an example, we present a detailed numerical study of the rates of convergence in terms of powers of ε for the error u⁰-u_h^eu^{0}-u_{h}^{\varepsilon}, and numerically examine what is the “best” mesh size h in relation to ε in order to achieve these rates.