Analysis of FETI methods for multiscale PDEs

In this paper, we study a variant of the finite element tearing and interconnecting (FETI) method which is suitable for elliptic PDEs with highly heterogeneous (multiscale) coefficients α(x); in particular, coefficients with strong variation within subdomains and/or jumps that are not aligned with the subdomain interfaces. Using energy minimisation and cut-off arguments we can show rigorously that for an arbitrary (positive) coefficient function the condition number of the preconditioned FETI system can be bounded by C(α) (1 + log(H/h))² where H is the subdomain diameter and h is the mesh size, and where the function C(α) depends only on the coefficient variation in the vicinity of subdomain interfaces. In particular, if varies only mildly in a layer Ω _i,η of width η near the boundary of each of the subdomains Ω _i , then , independent of the variation of α in the remainder Ω _i \Ω _i,η of each subdomain and independent of any jumps of α across subdomain interfaces. The quadratic dependence of C(α) on H/η can be relaxed to a linear dependence under stronger assumptions on the behaviour of α in the interior of the subdomains. Our theoretical findings are confirmed in numerical tests. C. Pechstein was supported by the Austrian Science Funds (FWF) under grant F1306.     

<Emphasis Type="Italic">hp</Emphasis>-Discontinuous Galerkin methods for strongly nonlinear elliptic boundary value problems

In this paper, we have analyzed a one parameter family of hp-discontinuous Galerkin methods for strongly nonlinear elliptic boundary value problems with Dirichlet boundary conditions. These methods depend on the values of the parameter , where θ = + 1 corresponds to the nonsymmetric and θ = −1 corresponds to the symmetric interior penalty methods when and f(u,∇u) = −f, that is, for the Poisson problem. The error estimate in the broken H ¹ norm, which is optimal in h (mesh size) and suboptimal in p (degree of approximation) is derived using piecewise polynomials of degree p ≥ 2, when the solution . In the case of linear elliptic problems also, this estimate is optimal in h and suboptimal in p. Further, optimal error estimate in the L ² norm when θ = −1 is derived. Numerical experiments are presented to illustrate the theoretical results. Supported by DST-DAAD (PPP-05) project.     

Fast linear algebra is stable

In Demmel et al. (Numer. Math. 106(2), 199–224, 2007) we showed that a large class of fast recursive matrix multiplication algorithms is stable in a normwise sense, and that in fact if multiplication of n-by-n matrices can be done by any algorithm in O(n ^ω+η ) operations for any η >  0, then it can be done stably in O(n ^ω+η ) operations for any η >  0. Here we extend this result to show that essentially all standard linear algebra operations, including LU decomposition, QR decomposition, linear equation solving, matrix inversion, solving least squares problems, (generalized) eigenvalue problems and the singular value decomposition can also be done stably (in a normwise sense) in O(n ^ω+η ) operations. J. Demmel acknowledges support of NSF under grants CCF-0444486, ACI-00090127, CNS-0325873 and of DOE under grant DE-FC02-01ER25478.     

General proximal point algorithm involving η-maximal accretiveness framework in Banach spaces

First, based on η-maximal accretiveness, a generalization to Rockafellar’s theorem (1976) in the context of approximating a solution to a general inclusion problem involving a multivalued η-maximal accretive mapping using the proximal point algorithm in a q-uniformly smooth Banach space setting is considered. Then an application to a minimization problem of a functional is examined. The general framework for η-maximal accretiveness generalizes the general theory of multivalued maximal monotone mappings.      

Saddle-Point Criteria in an η-Approximation Method for Nonlinear Mathematical Programming Problems Involving Invex Functions

In this paper, the η-approximation method introduced by Antczak (Ref. 1) for solving a nonlinear constrained mathematical programming problem involving invex functions with respect to the same function η is extended. In this method, a so-called η-approximated optimization problem associated with the original mathematical programming problems is constructed; moreover, an η-saddle point and an η-Lagrange function are defined. By the help of the constructed η-approximated optimization problem, saddle-point criteria are obtained for the original mathematical programming problem. The equivalence between an η-saddle point of the η-Lagrangian of the associated η-approximated optimization problem and an optimal solution in the original mathematical programming problem is established.     

A proximal point algorithm for minimax problems

This paper presents a proximal point algorithm for solving discretel _∞ approximation problems of the form minimize ∥Ax−b∥_∞. Let ε_∞ be a preassigned positive constant and let ε _l ,l = 0,1,2,... be a sequence of positive real numbers such that 0 < ε _l < ε_∞. Then, starting from an arbitrary pointz ₀, the proposed method generates a sequence of points z _l ,l= 0,1,2,..., via the rule . One feature that characterizes this algorithm is its finite termination property. That is, a solution is reached within a finite number of iterations. The smaller are the numbers ε _l the smaller is the number of iterations. In fact, if ε ₀ is sufficiently small then z₁ solves the original minimax problem. The practical value of the proposed iteration depends on the availability of an efficient code for solving a regularized minimax problem of the form minimize where ∈ is a given positive constant. It is shown that the dual of this problem has the form maximize , and ify solves the dual thenx=A ^T y solves the primal. The simple structure of the dual enables us to apply a wide range of methods. In this paper we design and analyze a row relaxation method which is suitable for solving large sparse problems. Numerical experiments illustrate the feasibility of our ideas.     

Robust multigrid method for the planar linear elasticity problems

We consider the solution of the system of equations that arise from the higher order conforming finite element (Scott–Vogelius element) discretizations of the boundary value problems associated with the differential operator −ρ ² Δ − κ ²∇div, where ρ and κ are nonzero parameters. Robust multigrid method is constructed, i.e., the convergence rate of multigrid method is optimal with respect to the mesh size, the number of levels, and weights on the two terms in the aforementioned differential operator.

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Second order duality for minmax fractional programming

In the present paper, two types of second order dual models are formulated for a minmax fractional programming problem. The concept of η-bonvexity/generalized η-bonvexity is adopted in order to discuss weak, strong and strict converse duality theorems. The research of Z. Husain is supported by the Department of Atomic Energy, Government of India, under the NBHM Post-Doctoral Fellowship Program No. 40/9/2005-R&D II/1739.     

Interpolation error estimates in W 1,p for degenerate Q 1 isoparametric elements

Optimal order error estimates in H ¹, for the Q ₁ isoparametric interpolation were obtained in Acosta and Durán (SIAM J Numer Anal37, 18–36, 1999) for a very general class of degenerate convex quadrilateral elements. In this work we show that the same conlusions are valid in W ^1,p for 1≤ p < 3 and we give a counterexample for the case p ≥ 3, showing that the result cannot be generalized for more regular functions. Despite this fact, we show that optimal order error estimates are valid for any p ≥ 1, keeping the interior angles of the element bounded away from 0 and π, independently of the aspect ratio. We also show that the restriction on the maximum angle is sharp for p ≥ 3.     

Auxiliary space preconditioning in H 0(curl; Ω)

We adapt the principle of auxiliary space preconditioning as presented in [J. Xu, The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids, Computing, 56 (1996), pp. 215–235.] to H (curl; ω)-elliptic variational problems discretized by means of edge elements. The focus is on theoretical analysis within the abstract framework of subspace correction. Employing a Helmholtz-type splitting of edge element vector fields we can establish asymptotic h-uniform optimality of the preconditioner defined by our auxiliary space method. This author was fully supported by Hong Kong RGC grant (Project No. 403403) This author acknowledges the support from a Direct Grant of CUHK during his visit at The Chinese University of Hong Kong.     

A convergent adaptive finite element method for an optimal design problem

The optimal design problem for maximal torsion stiffness of an infinite bar of given geometry and unknown distribution of two materials of prescribed amounts is one model example in topology optimisation. It eventually leads to a degenerate convex minimisation problem. The numerical analysis is therefore delicate for possibly multiple primal variables u but unique derivatives σ : = DW(D u). Even fine a posteriori error estimates still suffer from the reliability-efficiency gap. However, it motivates a simple edge-based adaptive mesh-refining algorithm (AFEM) that is not a priori guaranteed to refine everywhere. Its convergence proof is therefore based on energy estimates and some refined convexity control. Numerical experiments illustrate even nearly optimal convergence rates of the proposed AFEM. Supported by the DFG Research Center MATHEON “Mathematics for key technologies” in Berlin.     

Error analysis of finite element approximations of the inverse mean curvature flow arising from the general relativity

This paper proposes and analyzes a finite element method for a nonlinear singular elliptic equation arising from the black hole theory in the general relativity. The nonlinear equation, which was derived and analyzed by Huisken and Ilmanen in (J Diff Geom 59:353–437), represents a level set formulation for the inverse mean curvature flow describing the evolution of a hypersurface whose normal velocity equals the reciprocal of its mean curvature. We first propose a finite element method for a regularized flow which involves a small parameter ɛ; a rigorous analysis is presented to study well-posedness and convergence of the scheme under certain mesh-constraints, and optimal rates of convergence are verified. We then prove uniform convergence of the finite element solution to the unique weak solution of the nonlinear singular elliptic equation as the mesh size h and the regularization parameter ɛ both tend to zero. Computational results are provided to show the efficiency of the proposed finite element method and to numerically validate the “jumping out” phenomenon of the weak solution of the inverse mean curvature flow. Numerical studies are presented to evidence the existence of a polynomial scaling law between the mesh size h and the regularization parameter ɛ for optimal convergence of the proposed scheme. Finally, a numerical convergence study for another approach recently proposed by R. Moser (The inverse mean curvature flow and p-harmonic functions. preprint U Bath, 2005) for approximating the inverse mean curvature flow via p-harmonic functions is also included.     

Inf-sup stable non-conforming finite elements of arbitrary order on triangles

We introduce a family of scalar non-conforming finite elements of arbitrary order k≥1 with respect to the H¹-norm on triangles. Their vector-valued version generates together with a discontinuous pressure approximation of order k−1 an inf-sup stable finite element pair of order k for the Stokes problem in the energy norm. For k=1 the well-known Crouzeix-Raviart element is recovered.     

Adaptive finite element for semi-linear convection–diffusion problems

In this article a strategy of adaptive finite element for semi-linear problems, based on minimizing a residual-type estimator, is reported. We get an a posteriori error estimate which is asymptotically exact when the mesh size h tends to zero. By considering a model problem, the quality of this estimator is checked. It is numerically shown that without constraint on the mesh size h, the efficiency of the a posteriori error estimate can fail dramatically. This phenomenon is analysed and an algorithm which equidistributes the local estimators under the constraint h ⩽ h _max is proposed. This algorithm allows to improve the computed solution for semi-linear convection–diffusion problems, and can be used for stabilizing the Lagrange finite element method for linear convection–diffusion problems. This revised version was published online in June 2006 with corrections to the Cover Date.     

Adaptive eigenvalue computation: complexity estimates

This paper is concerned with the design and analysis of a fully adaptive eigenvalue solver for linear symmetric operators. After transforming the original problem into an equivalent one formulated on ℓ ₂, the space of square summable sequences, the problem becomes sufficiently well conditioned so that a gradient type iteration can be shown to reduce the error by some fixed factor per step. It then remains to realize these (ideal) iterations within suitable dynamically updated error tolerances. It is shown under which circumstances the adaptive scheme exhibits in some sense asymptotically optimal complexity. This research was supported in part by the Leibniz Programme of the DFG, by the SFB 401 funded by DFG, the DFG Priority Program SPP1145 and by the EU NEST project BigDFT.     

Multigrid methods for the computation of singular solutions and stress intensity factors II: Crack singularities

We consider the Poisson equation −Δu=f with homogeneous Dirichlet boundary condition on a two-dimensional polygonal domain Ω with cracks. Multigrid methods for the computation of singular solutions and stress intensity factors using piecewise linear functions are analyzed. The convergence rate for the stress intensity factors is whenfεL ²(Ω) and whenfεH ¹(Ω). The convergence rate in the energy norm is in the first case and in the second case. The costs of these multigrid methods are proportional to the number of elements in the triangulation. The general case wherefεH ^m (Ω) is also discussed. The work of the first author was partially supported by NSF under grant DMS-96-00133     

A trapping principle and convergence result for finite element approximate solutions of steady reaction/diffusion systems

We consider nonlinear elliptic systems, with mixed boundary conditions, on a convex polyhedral domain Ω ⊂ R ^N . These are nonlinear divergence form generalizations of Δu = f(·, u), where f is outward pointing on the trapping region boundary. The motivation is that of applications to steady-state reaction/diffusion systems. Also included are reaction/diffusion/convection systems which satisfy the Einstein relations, for which the Cole-Hopf transformation is possible. For maximum generality, the theory is not tied to any specific application. We are able to demonstrate a trapping principle for the piecewise linear Galerkin approximation, defined via a lumped integration hypothesis on integrals involving f, by use of variational inequalities. Results of this type have previously been obtained for parabolic systems by Estep, Larson, and Williams, and for nonlinear elliptic equations by Karátson and Korotov. Recent minimum and maximum principles have been obtained by Jüngel and Unterreiter for nonlinear elliptic equations. We make use of special properties of the element stiffness matrices, induced by a geometric constraint upon the simplicial decomposition. This constraint is known as the non-obtuseness condition. It states that the inward normals, associated with an arbitrary pair of an element’s faces, determine an angle with nonpositive cosine. Drăgănescu, Dupont, and Scott have constructed an example for which the discrete maximum principle fails if this condition is omitted. We also assume vertex communication in each element in the form of an irreducibility hypothesis on the off-diagonal elements of the stiffness matrix. There is a companion convergence result, which yields an existence theorem for the solution. This entails a consistency hypothesis for interpolation on the boundary, and depends on the Tabata construction of simple function approximation, based on barycentric regions. This work was supported by the National Science Foundation under grant DMS-0311263.     

Characteristics difference method for incompressible miscible flow in porous media

Two-phase ,incompressible miscible flow in porous media is governed by a system ofnonlinear partial differential equations. The pressure equation ,which is e11iptic in appearance ,isdiseretizod by a standard five-points difference method, The concentration equation is treated byan impliclt finite difference method that appbes a form of the method of characterlstics to thetransport terms. A class of biquadlatle interpolation is introduced for the method of chracteristics.Convergence rate is proved to be O(△t h^2)。     

Analysis of a BDF–DGFE scheme for nonlinear convection–diffusion problems

We deal with the numerical solution of a scalar nonstationary nonlinear convection–diffusion equation. We employ a combination of the discontinuous Galerkin finite element method for the space semi-discretization and the k-step backward difference formula for the time discretization. The diffusive and stabilization terms are treated implicitly whereas the nonlinear convective term is treated by a higher order explicit extrapolation method, which leads to the necessity to solve only a linear algebraic problem at each time step. We analyse this scheme and derive a priori asymptotic error estimates in the discrete L ^∞(L ²)-norm and the L ²(H ¹)-seminorm with respect to the mesh size h and time step τ for k = 2,3. Numerical examples verifying the theoretical results are presented. This work is a part of the research project MSM 0021620839 financed by the Ministry of Education of the Czech Republic and was partly supported by the Grant No. 316/2006/B-MAT/MFF of the Grant Agency of the Charles University Prague. The research of M. Vlasák was supported by the project LC06052 of the Ministry of Education of the Czech Republic (Jindřich Nečas Center for Mathematical Modelling).     

A second order <Emphasis Type="Italic">η</Emphasis>-approximation method for constrained optimization problems involving second order invex functions

A new approach for obtaining the second order sufficient conditions for non-linear mathematical programming problems which makes use of second order derivative is presented. In the so-called second order η-approximation method, an optimization problem associated with the original nonlinear programming problem is constructed that involves a second order η-approximation of both the objective function and the constraint function constituting the original problem. The equivalence between the nonlinear original mathematical programming problem and its associated second orderη-approximated optimization problem is established under second order invexity assumption imposed on the functions constituting the original optimization problem.