Inhomogeneous Dirichlet conditions in a priori and a posteriori finite element error analysis

Summary. The numerical solution of elliptic boundary value problems with finite element methods requires the approximation of given Dirichlet data u_D by functions u_D,h in the trace space of a finite element space on _D. In this paper, quantitative a priori and a posteriori estimates are presented for two choices of u_D,h, namely the nodal interpolation and the orthogonal projection in L²(_D) onto the trace space. Two corresponding extension operators allow for an estimate of the boundary data approximation in global H¹ and L² a priori and a posteriori error estimates. The results imply that the orthogonal projection leads to better estimates in the sense that the influence of the approximation error on the estimates is of higher order than for the nodal interpolation.Mathematics Subject Classification (1991): 65N30, 65R20, 73C50This work was initiated while C. Carstensen was visiting the Max Planck Institute for Mathematics in the Sciences, Leipzig. S. Bartels acknowledges support by the German Research Foundation (DFG) within the Graduiertenkolleg Effiziente Algorithmen und Mehrskalenmethoden and the priority program Analysis, Modeling, and Simulation of Multiscale Problems. G. Dolzmann gratefully acknowledges partial support by the Max Planck Society and by the NSF through grant DMS0104118.     

Boundary element monotone iteration scheme for semilinear elliptic partial differential equations

The monotone iteration scheme is a constructive method for solving a wide class of semilinear elliptic boundary value problems. With the availability of a supersolution and a subsolution, the iterates converge monotonically to one or two solutions of the nonlinear PDE. However, the rates of such monotone convergence cannot be determined in general. In addition, when the monotone iteration scheme is implemented numerically through the boundary element method, error estimates have not been analyzed in earlier studies. In this paper, we formulate a working assumption to obtain an exponentially fast rate of convergence. This allows a margin for the numerical implementation of boundary elements within the range of monotone convergence. We then interrelate several approximate solutions, and use the Aubin-Nitsche lemma and the triangle inequalities to derive error estimates for the Galerkin boundary-element iterates with respect to the , , Sobolev space norms. Such estimates are of optimal order. Furthermore, as a peculiarity, we show that for the nonlinearities that are of separable type, ``higher than optimal order' error estimates can be obtained with respect to the mesh parameter . Several examples of semilinear elliptic partial differential equations featuring different situations of existence/nonexistence, uniqueness/multiplicity and stability are discussed, computed, and the graphics of their numerical solutions are illustrated.

     

Regularity criteria in terms of pressure for the 3-D Navier-Stokes equations in a generic domain

We consider the 3-D Navier-Stokes equations in the half-space ₊³, or a bounded domain with smooth boundary, or else an exterior domain with smooth boundary. Some new sufficient conditions on pressure or the gradient of pressure for the regularity of weak solutions to the Navier-Stokes equations are obtained.Mathematics Subject Classification (2000):35B45, 35B65, 76D05     

Analysis of a fourth order finite difference method for the incompressible Boussinesq equations

Summary. The convergence of a fourth order finite difference method for the 2-D unsteady, viscous incompressible Boussinesq equations, based on the vorticity-stream function formulation, is established in this article. A compact fourth order scheme is used to discretize the momentum equation, and long-stencil fourth order operators are applied to discretize the temperature transport equation. A local vorticity boundary condition is used to enforce the no-slip boundary condition for the velocity. One-sided extrapolation is used near the boundary, dependent on the type of boundary condition for the temperature, to prescribe the temperature at ghost points lying outside of the computational domain. Theoretical results of the stability and accuracy of the method are also provided. In numerical experiments the method has been shown to be capable of producing highly resolved solutions at a reasonable computational cost.Mathematics Subject Classification (1991): 35Q35, 65M06, 76M20     

Recovery of An Unknown Flux in Parabolic Problems with Nonstandard Boundary Conditions: Error Estimates

In this paper, we consider a 2nd order semilinear parabolic initial boundary value problem (IBVP) on a bounded domain ^N, with nonstandard boundary conditions (BCs). More precisely, at some part of the boundary we impose a Neumann BC containing an unknown additive space-constant (t), accompanied with a nonlocal (integral) Dirichlet side condition.We design a numerical scheme for the approximation of a weak solution to the IBVP and derive error estimates for the approximation of the solution u and also of the unknown function .     

An adaptive strategy for elliptic problems including a posteriori controlled boundary approximation

We derive a posteriori error estimates for the approximation of linear elliptic problems on domains with piecewise smooth boundary. The numerical solution is assumed to be defined on a Finite Element mesh, whose boundary vertices are located on the boundary of the continuous problem. No assumption is made on a geometrically fitting shape.

A posteriori error estimates are given in the energy norm and the -norm, and efficiency of the adaptive algorithm is proved in the case of a saturated boundary approximation. Furthermore, a strategy is presented to compute the effect of the non-discretized part of the domain on the error starting from a coarse mesh. This especially implies that parts of the domain, where the measured error is small, stay non-discretized. The presented algorithm includes a stable path following to supply a sufficient polygonal approximation of the boundary, the reliable computation of the a posteriori estimates and a mesh adaptation based on Delaunay techniques. Numerical examples illustrate that errors outside the initial discretization will be detected.

     

Numerical solution of a time-like Cauchy problem for the wave equation

Let D ? ?ⁿ be a bounded domain with piecewise-smooth boundary, and q(x,t) a smooth function on D × [0, T]. Consider the time-like Cauchy problem Given g, h for which the equation has a solution, we show how to approximate u(x,t) by solving a well posed fourth-order elliptic partial differential equation (PDE). We use the method of quasi-reversibility to construct the approximating PDE. We derive error estimates and present numerical results.     

Boundary Element Methods for Maxwell Transmission Problems in Lipschitz Domains

Summary. We consider the Maxwell equations in a domain with Lipschitz boundary and the boundary integral operator A occuring in the Calderón projector. We prove an inf-sup condition for A using a Hodge decomposition. We apply this to two types of boundary value problems: the exterior scattering problem by a perfectly conducting body, and the dielectric problem with two different materials in the interior and exterior domain. In both cases we obtain an equivalent boundary equation which has a unique solution. We then consider Galerkin discretizations with Raviart-Thomas spaces. We show that these spaces have discrete Hodge decompositions which are in some sense close to the continuous Hodge decomposition. This property allows us to prove quasioptimal convergence of the resulting boundary element methods. Mathematics Subject Classification (2000):65N30     

Finite element approximation of spectral acoustic problems on curved domains

Summary This paper deals with the finite element approximation of the displacement formulation of the spectral acoustic problem on a curved non convex two-dimensional domain . Convergence and error estimates are proved for Raviart-Thomas elements on a discrete polygonal domain _h in the framework of the abstract spectral approximation theory. Similar results have been previously proved only for polygonal domains. Numerical tests confirming the theoretical results are reported.Mathematics Subject Classification (2000):65N25, 65N30, 70J30Supported by FONDECYT 2000114 (Chile)Supported by FONDAP in Applied Mathematics (Chile)     

Decay Properties of Klein‐Gordon Fields on Kerr‐AdS Spacetimes

This paper investigates the decay properties of solutions to the massive linear wave equation for g being the metric of a Kerr‐AdS spacetime satisfying satisfying the Breitenlohner‐Freedman bound. We prove that the nondegenerate energy of ψ with respect to an appropriate foliation of spacelike slices decays like (log t^?)^?2. Our estimates are expected to be sharp from heuristic and numerical arguments in the physics literature suggesting that general solutions will only decay logarithmically. The underlying reason for the slow decay rate can be traced back to a stable trapping phenomenon for asymptotically anti‐de Sitter black holes, which is in turn a consequence of the reflecting boundary conditions at null infinity.© 2013 Wiley Periodicals, Inc.     

An Integral Geometry Problem in a Nonconvex Domain

We consider the problem of recovering the solenoidal part of a symmetric tensor field f on a compact Riemannian manifold (M,g) with boundary from the integrals of f over all geodesics joining boundary points. All previous results on the problem are obtained under the assumption that the boundary M is convex. This assumption is related to the fact that the family of maximal geodesics has the structure of a smooth manifold if M is convex and there is no geodesic of infinite length in M. This implies that the ray transform of a smooth field is a smooth function and so we may use analytic techniques. Instead of convexity of M we assume that M is a smooth domain in a larger Riemannian manifold with convex boundary and the problem under consideration admits a stability estimate. We then prove uniqueness of a solution to the problem for     

On the Multiplicity of Solutions Of a Differential Equation Arising in Chemical Reactor Theory

Consider the boundary value problem where β ? 0, τ ? 0. We are concerned with a mathematically rigorous numerical study of the number of solutions in any bounded portion of the positive quadrant (τ ? 0, β ? 0) of the τ, β plane. These correct computational results may then be matched with asymptotic (β→∞, τ ? 0) results developed earlier. These numerical results are based on the development of a posteriori error estimates for the numerical solution of an associated initial-value problem and a priori bounds on .     

A posteriori error estimates and domain decomposition with nonmatching grids

Let F be a nonlinear mapping defined from a Hilbert space X into its dual X, and let x be in X the solution of F(x)=0. Assume that, a priori, the zone where the gradient of the function x has a large variation is known. The aim of this article is to prove a posteriori error estimates for the problem F(x)=0 when it is approximated with a Petrov–Galerkin finite element method combined with a domain decomposition method with nonmatching grids. A residual estimator for a model semi-linear problem is proposed. We prove that this estimator is asymptotically equivalent to a simplified one adapted to parallel computing. Some numerical results are presented, showing the practical efficiency of the estimator. AMS subject classification 65J10, 65N55, 65M60T. Sassi: Present address: Université de Caen, Laboratoire de Mathématiques Nicolas Oresme, UFR Sciences Campus II, Bd Maréchal Juin, 14032 Caen Cedex, France.     

Hearing pseudoconvexity with the Kohn Laplacian

A bounded domain in with connected Lipschitz boundary is pseudoconvex if the bottom of the essential spectrum of the Kohn Laplacian on the space of (0,q)-forms, 1qn–1, with L²-coefficients is positive.The author was supported in part by NSF grant DMS 0070697 and by an AMS centennial fellowship.Revised version: 9 July 2004     

Error estimates for the numerical approximation of time-dependent flow of Bingham fluid in cylindrical pipes by the regularization method

Summary The flow of a Bingham fluid in a cylindrical pipe can give rise to free boundary problems. The fluid behaves like a viscous fluid if the shear stress, expressed as a linear function of the shear rate, exceeds a yield value, and like a rigid body otherwise. The surfaces dividing fluid and rigid zones are the free boundaries. Therefore the solution for such highly nonlinear problems can in general only be obtained by numerical methods. Considerable progress has been made in the development of numerical algorithms for Bingham fluids [2,20,27,29,30,32]. However, very little research can be found in the literature regarding the rate of convergence of the numerical solution to the true continuous solution, that is the error estimate of these numerical methods. Error estimates are a critically important issue because they tells us how to control the error by appropriately choosing the grid sizes and other related parameters. This paper concerns the error estimates of a unsteady Bingham fluid modeled as a variational inequality due to Duvaut-Lions [16] and Glowinski [30]. The difficulty both in the analytical and numerical treatment of the mathematical model is due to the fact that it contains a nondifferentiable term. A common technique, called the regularization method, is to replace the non-differentiable term by a perturbed differentiable term which depends on a small regularization parameter . The regularization method effectively reduces the variational inequality to an equation (a regularized problem) which is much easier to cope with. This paper has achieved the following. (1) Error estimates are derived for a continuous time Galerkin method in suitable norms. (2) We give an estimate of the difference between the true solution and the regularized solution in terms of . (3) Some regularity properties for both regularized solution and the true solution are proved. (4) The error estimates for full discretization of the regularized problem using piecewise linear finite elements in space, and backward differencing in time are established for the first time by coupling the regularization parameter and the discretization parameters h and t. (5) We are able to improve our estimates in the one-dimensional case or under stronger regularity assumptions on the true solution. The estimates for the one-dimensional case are optimal and confirmed by numerical experiment. The estimates from (4) and (5) provide very important information on the measure of the error and give us a powerful mechanism to properly choose the parameters h, t and in order to control the error (see Corollary 4.4). The above estimates extend the error bounds derived in Glowinski, Lions and Trémolières [32] (chapter 5, pp. 348–404) for the stationary Bingham fluid to the time-dependent one, which is the main contribution of this paper. Mathematics Subject Classification (2000):35k85, 65M15, 65M60, 76A05, 76M10I wish to thank my thesis advisor, Professor Todd Dupont, for his motivation and help on the writing of this paper during my Ph.D study at the University of Chicago. I also want to thank my postdoctoral advisor, Professor James Glimm, for his assistance with improvements to this paper.     

A priori and a posteriori analysis of finite volume discretizations of Darcy’s equations

Summary This paper is devoted to the numerical analysis of some finite volume discretizations of Darcys equations. We propose two finite volume schemes on unstructured meshes and prove their equivalence with either conforming or nonconforming finite element discrete problems. This leads to optimal a priori error estimates. In view of mesh adaptivity, we exhibit residual type error indicators and prove estimates which allow to compare them with the error in a very accurate way. Mathematics Subject Classification (2000):65G99, 65M06, 65M15, 65M60, 65P05This work was partially supported by Contract C03127/AEE2714 with the Laboratoire National dHydraulique of the Division Recherche et Développement of Électricité de France. We thank B. Gest and her research group for very interesting discussions on this subject.     

Exit probabilities and optimal stochastic control

This paper is concerned with Markov diffusion processes which obey stochastic differential equations depending on a small parameter. The parameter enters as a coefficient in the noise term of the stochastic differential equation. The Ventcel-Freidlin estimates give asymptotic formulas (as0) for such quantities as the probability of exit from a regionD through a given portionN of the boundary D, the mean exit time, and the probability of exit by a given timeT. A new method to obtain such estimates is given, using ideas from stochastic control theory.This research was supported by the Air Force Office of Scientific Research under AF-AFOSR 76-3063, and in part by the National Science Foundation under NSF-MCS 76-37247.     

Adaptive Galerkin boundary element methods with panel clustering

In this paper, we will propose a boundary element method for solving classical boundary integral equations on complicated surfaces which, possibly, contain a large number of geometric details or even uncertainties in the given data. The (small) size of such details is characterised by a small parameter and the regularity of the solution is expected to be low in such zones on the surface (which we call the wire-basket zones). We will propose the construction of an initial discretisation for such type of problems. Afterwards standard strategies for boundary element discretisations can be applied such as the h, p, and the adaptive hp-version in a straightforward way. For the classical boundary integral equations, we will prove the optimal approximation results of our so-called wire-basket boundary element method and discuss the stability aspects. Then, we construct the panel-clustering and -matrix approximations to the corresponding Galerkin BEM stiffness matrix. The method is shown to have an almost linear complexity with respect to the number of degrees of freedom located on the wire basket.     

Discrete minimum and maximum principles for finite element approximations of non-monotone elliptic equations

Summary. Uniform lower and upper bounds for positive finite-element approximations to semilinear elliptic equations in several space dimensions subject to mixed Dirichlet-Neumann boundary conditions are derived. The main feature is that the non-linearity may be non-monotone and unbounded. The discrete minimum principle provides a positivity-preserving approximation if the discretization parameter is small enough and if some structure conditions on the non-linearity and the triangulation are assumed. The discrete maximum principle also holds for degenerate diffusion coefficients. The proofs are based on Stampacchias truncation technique and on a variational formulation. Both methods are settled on careful estimates on the truncation operator.Mathematics Subject Classification (2000): 65N30, 65N12     

Analysis of a cell boundary element method

The CBEM (cell boundary element method) was proposed as a numerical method for second-order elliptic problems by the first author in the earlier paper [10]. In this paper we prove a quasi-optimal order of convergence of the method, O(h^1–) for >0 in H¹-norm for the triangular mesh; also a stability result is obtained. We provide numerical examples and it is observed that the method conserves flux exactly when a certain condition on meshes is satisfied. This work was supported by KOSEF 2000-1-10300-001-5.AMS subject classification 65N30, 65N38, 65N50