General Lipschitz Research Articles

Is the one‐equation coupling of finite and boundary element methods always stable?

AbstractIn this paper we present a sufficient and necessary condition to ensure the ellipticity of the bilinear form which is related to the one‐equation coupling of finite and boundary element methods to solve a scalar free space transmission problem for a second order uniform elliptic partial differential equation in the case of general Lipschitz interfaces. This condition relates the minimal eigenvalue of the coefficient matrix in the bounded interior domain to the contraction constant of the shifted double layer integral operator which reflects the shape of the interface. This paper extends and improves earlier results [12] on sufficient conditions, but now includes also necessary conditions. Numerical examples confirm the theoretical results on the sharpeness of the presented estimates.

Toward structural sparsity: an explicit $$\ell _{2}/\ell _0$$ approach

As powerful tools, machine learning and data mining techniques have been widely applied in various areas. However, in many real-world applications, besides establishing accurate black box predictors, we are also interested in white box mechanisms, such as discovering predictive patterns in data that enhance our understanding of underlying physical, biological and other natural processes. For these purposes, sparse representation and its variations have been one of the focuses. More recently, structural sparsity has attracted increasing attentions. In previous research, structural sparsity was often achieved by imposing convex but non-smooth norms such as $${\ell _{2}/\ell _{1}}$$ and group $${\ell _{2}/\ell _{1}}$$ norms. In this paper, we present the explicit $${\ell _2/\ell _0}$$ and group $${\ell _2/\ell _0}$$ norm to directly approach the structural sparsity. To tackle the problem of intractable $${\ell _2/\ell _0}$$ optimizations, we develop a general Lipschitz auxiliary function that leads to simple iterative algorithms. In each iteration, optimal solution is achieved for the induced subproblem and a guarantee of convergence is provided. Furthermore, the local convergent rate is also theoretically bounded. We test our optimization techniques in the multitask feature learning problem. Experimental results suggest that our approaches outperform other approaches in both synthetic and real-world data sets.

Approximations of fractional stochastic differential equations by means of transport processes

We present strong approximations with rate of convergence for the solution of a stochastic differential equation of the form $$ dX_t=b(X_t)dt+\sigma(X_t)dB^H_t, $$ where $b\in C^1_b$, $\sigma \in C^2_b$, $B^H$ is fractional Brownian motion with Hurst index $H$, and we assume existence of a unique solution with Doss-Sussmann representation. The results are based on a strong approximation of $B^H$ by means of transport processes of Garz\'on et al (2009). If $\sigma$ is bounded away from 0, an approximation is obtained by a general Lipschitz dependence result of R\"omisch and Wakolbinger (1985). Without that assumption on $\sigma$, that method does not work, and we proceed by means of Euler schemes on the Doss-Sussmann representation to obtain another approximation, whose proof is the bulk of the paper.

A Note on the Stable One-Equation Coupling of Finite and Boundary Elements

In a recent paper [SIAM J. Numer. Anal., 47 (2009), pp. 3451–3463] Sayas proved the stability of the Johnson–Nédélec coupling of finite and boundary element methods on polygonal interfaces when the direct boundary integral equation with single and double layer integral operators is used only. In this note we present two alternative proofs of this result for general Lipschitz interfaces. In particular, we prove an ellipticity estimate of the coupled bilinear form. Hence, we can use standard arguments to derive stability and error estimates for the Galerkin discretization for all pairs of finite and boundary element trial spaces.

Nonsymmetric coupling of BEM and mixed FEM on polyhedral interfaces

In this paper we propose and analyze some new methods for coupling mixed finite element and boundary element methods for the model problem of the Laplace equation in free space or in the exterior of a bounded domain. As opposed to the existing methods, which use the complete matrix of operators of the Calderon projector to obtain a symmetric coupled system, we propose methods with only one integral equation. The system can be considered as a further generalization of the Johnson-Nedelec coupling of BEM—FEM to the case of mixed formulations in the bounded domain. Using some recent analytical tools we are able to prove stability and convergence of Galerkin methods with very general conditions on the discrete spaces and no restriction relating the finite and boundary element spaces. This can be done for general Lipschitz interfaces and in particular, the coupling boundary can be taken to be a Lipschitz polyhedron. Both the indirect and the direct approaches for the boundary integral formulation are explored.

A mixed finite element method with exactly divergence-free velocities for incompressible magnetohydrodynamics

We introduce and analyze a mixed finite element method for the numerical discretization of a stationary incompressible magnetohydrodynamics problem, in two and three dimensions. The velocity field is discretized using divergence-conforming Brezzi–Douglas–Marini (BDM) elements and the magnetic field is approximated by curl-conforming Nédélec elements. The H 1-continuity of the velocity field is enforced by a DG approach. A central feature of the method is that it produces exactly divergence-free velocity approximations, and captures the strongest magnetic singularities. We prove that the energy norm error is convergent in the mesh size in general Lipschitz polyhedra under minimal regularity assumptions, and derive nearly optimal a priori error estimates for the two-dimensional case. We present a comprehensive set of numerical experiments, which indicate optimal convergence of the proposed method for two-dimensional as well as three-dimensional problems.

Iterative Algorithms with Variable Coefficients for Multivalued Generalized -Hemicontractive Mappings without Generalized Lipschitz Assumption

We introduce and study some new Ishikawa-type iterative algorithms with variable coefficients for multivalued generalized Open image in new window-hemicontractive mappings. Several new fixed-point theorems for multivalued generalized Open image in new window-hemicontractive mappings without generalized Lipschitz assumption are established in Open image in new window-uniformly smooth real Banach spaces. A result for multivalued generalized Open image in new window-hemicontractive mappings with bounded range is obtained in uniformly smooth real Banach spaces. As applications, several theorems for multivalued generalized Open image in new window-hemiaccretive mapping equations are given.

Optimal Error Estimates for Nédélec Edge Elements for Time-Harmonic Maxwell's Equations

In this paper, we obtain optimal error estimates in both L 2 -norm and H(curl)-norm for the Nedelec edge flnite element approximation of the time-harmonic Maxwell's equations on a general Lipschitz domain discretized on quasi-uniform meshes. One key to our proof is to transform the L 2 error estimates into the L 2 estimate of a discrete divergence-free function which belongs to the edge flnite element spaces, and then use the approximation of the discrete divergence-free function by the continuous divergence-free function and a duality argument for the continuous divergence-free function. For Nedelec's second type elements, we present an optimal convergence estimate which improves the best results available in the literature.

Hodge decomposition for symmetric matrix fields and the elasticity complex in Lipschitz domains

In 1999 M. Eastwood has used the general construction known as the Bernstein-Gelfand-Gelfand (BGG) resolution to prove, at least in smooth situation, the equivalence of the linear elasticity complex and of the de Rham complex in $\mathbf{R}^{3}$. The main objective of this paper is to study the linear elasticity complex for general Lipschitz domains in $\mathbf{R}^{3}$ and deduce a complete Hodge orthogonal decomposition for symmetric matrix fields in $L^{2}$, counterpart of the Hodge decomposition for vector fields. As a byproduct one obtains that the finite dimensional terms of this Hodge decomposition can be interpreted in homological terms as the corresponding terms for the de Rham complex if one takes the homology with value in $RIG\cong \mathbf{R}^{6}$ as in the (BGG) resolution.

A Galerkin Boundary Element Method for High Frequency Scattering by Convex Polygons

In this paper we consider the problem of time‐harmonic acoustic scattering in two dimensions by convex polygons. Standard boundary or finite element methods for acoustic scattering problems have a computational cost that grows at least linearly as a function of the frequency of the incident wave. Here we present a novel Galerkin boundary element method, which uses an approximation space consisting of the products of plane waves with piecewise polynomials supported on a graded mesh, with smaller elements closer to the corners of the polygon. We prove that the best approximation from the approximation space requires a number of degrees of freedom to achieve a prescribed level of accuracy that grows only logarithmically as a function of the frequency. Numerical results demonstrate the same logarithmic dependence on the frequency for the Galerkin method solution. Our boundary element method is a discretization of a well‐known second kind combined‐layer‐potential integral equation. We provide a proof that this equation and its adjoint are well‐posed and equivalent to the boundary value problem in a Sobolev space setting for general Lipschitz domains.

Lipschitz Domains, Domains with Corners, and the Hodge Laplacian

ABSTRACT We define self-adjoint extensions of the Hodge Laplacian on Lipschitz domains in Riemannian manifolds, corresponding to either the absolute or the relative boundary condition, and examine regularity properties of these operators' domains and form domains. We obtain results valid for general Lipschitz domains, and stronger results for a special class of “almost convex” domains, which apply to domains with corners.

Further results on the existence, uniqueness and stability of strong solutions of quantum stochastic differential equations

Under a more general Lipschitz condition on the coefficients than our consideration in [E.O. Ayoola, Existence and stability results for strong solutions of quantum stochastic differential equations, Stochastic Anal. Appl. 20 (2) (2002) 263–281], we establish the existence, uniqueness and stability of strong solutions of quantum stochastic differential equations (QSDE). This enables us to exhibit a class of Lipschitzian QSDE whose coefficients are continuous on the locally convex space of solution.

Convergence of the Newton method and uniqueness of zeros of vector fields on Riemannian manifolds

The estimates of the radii of convergence balls of the Newton method and uniqueness balls of zeroes of vector fields on the Riemannian manifolds are given under the assumption that the covariant derivatives of the vector fields satisfy some kind of general Lipschitz conditions. Some classical results such as the Kantorovich's type theorem and the Smale's γ -theory are extended.

A unifying local–semilocal convergence analysis and applications for two-point Newton-like methods in Banach space

We provide a local as well as a semilocal convergence analysis for two-point Newton-like methods in a Banach space setting under very general Lipschitz type conditions. Our equation contains a Fréchet differentiable operator F and another operator G whose differentiability is not assumed. Using more precise majorizing sequences than before we provide sufficient convergence conditions for Newton-like methods to a locally unique solution of equation F( x)+ G( x)=0. In the semilocal case we show under weaker conditions that our error estimates on the distances involved are finer and the information on the location of the solution at least as precise as in earlier results. In the local case a larger radius of convergence is obtained. Several numerical examples are provided to show that our results compare favorably with earlier ones. As a special case we show that the famous Newton–Kantorovich hypothesis is weakened under the same hypotheses as the ones contained in the Newton–Kantorovich theorem.

Multiscale asymptotic expansion and a post-processing algorithm for second-order elliptic problems with highly oscillatory coefficients over general convex domains

In this paper, we consider the boundary value problem of second-order elliptic type equation with highly oscillatory coefficients over general Lipschitz convex domains, and obtain the multiscale asymptotic expansion of solution for this kind of problem. At the same time, a high accuracy FE computing scheme and a post-processing technique are presented. Finally, numerical results support strongly the theoretical analysis reported in this paper.

Approximate controllability and regularity for semilinear retarded control systems

We deal with the approximate controllability for semilinear systems with time delay in a Hilbert space. First, we show the existence and uniqueness of solutions of the given systems with the more general Lipschitz continuity of nonlinear operator f from R × V to H. Thereafter, it is shown that the equivalence between the reachable set of the semilinear system and that of its corresponding linear system. Finally, we make a practical application of the conditions to the system with only discrete delay.

Nonlinear Equations with Operators Satisfying Generalized Lipschitz Conditions in Scales

By means of the contraction principle we prove existence, uniqueness and stability of solutions for nonlinear equations u + G_0[D, tL] + L(G_1 [D, u], G2[D, u]) = f in a Banach space E , where G_0, G_1 , G_2 satisfy Lipschitz conditions in scales of norms, L is a bilinear operator and D is a data parameter. The theory is applicable for inverse problems of memory identification and generalized convolution equations of the second kind.

Fully discrete finite element approaches for time-dependent Maxwell's equations

A fully discrete finite element method is used to approximate the electric field equation derived from time-dependent Maxwell's equations in three dimensional polyhedral domains. Optimal energy-norm error estimates are achieved for general Lipschitz polyhedral domains. Optimal L2 -norm error estimates are obtained for convex polyhedral domains.

Duality for General Lipschitz Classes and Applications

As shown by the author in {\em Proc. Amer. Math. Soc.} 115 (1992) 345–352, for every metric space $(\mbox{lip}\,\varphi(K))^{**} = \mbox{Lip} \,\varphi (K),$ is any majorant (that is, non-decreasing function on ) such that Here with respect to the metric is the corresponding ’little‘ Lipschitz space of functions vanishing ’at infinity‘, and ’=‘ means ’canonically isometrically isomorphic‘. The main idea of the proof consisted of finding a normed space $M^c = (\mbox{lip}\,\varphi(K))^*,$‘ stands for the completion, and identifying equipped with the Kantorovich norm. In the present paper, this argument is carried over to generalized Lipschitz spaces on defined in terms of higher order differences. For an integer $\lim_{t \rightarrow 0}\varphi(t)/t ^k = +\infty,$ to be the space of all bounded functions such that for some constant , where $\lambda ^k_\varphi$ which consists of functions vanishing at ’infinity' and such that We introduce an appropriate analogue of the Kantorovich norm on the space with compact support. The properties of this norm for are significantly different from those of the Kantorovich norm () which reflects the difference in analytic nature of generalized Lipschitz spaces versus classical ones. However, the core of the duality theory survives, and it is shown that $(M, \Vert\cdot \Vert _{k, \varphi})^* = \Lambda ^k _\varphi, (M, \Vert \cdot \Vert _{k,\varphi})^c = (\lambda ^k _\varphi)^*(\lambda ^k _\varphi )^{**} = \Lambda ^k _\varphi$. Several applications of these results are discussed, and a few open problems are formulated. 1991 Mathematics Subject Classification: 46E15, 46E35.

The sharpness of a pointwise error bound in connection with linear finite elements

Given a variational formulation of the two-dimensional Poisson problem, the present paper studies the error in approximating the exact solution via the finite element method baised upon linear elements. Using K-functional techniques, a direct sup-norm estimate in terms of a modulus of continuity of the exact solution follows directly from a well-known Jackson-type result. Because of the singularity of the Green's function associated the error bound contains a typical logarithm factor. Indeed, R. Haverkamp (1984) hats shown that this log-factor is necessary. His proof employs properties of the discrete Green's function for a related five point difference discretization. We adopt this method to establish the pointwise sharpness of the intermediate error estimate on a dense set in connection with general Lipschitz classes. The basic tool additionally used is a quantitative extension of the uniform boundedness principle. Mathematics Subject Classifications 1991: 41A25, 65N15, 65N30