On a method for studying the norm and the stability of solutions

We present a new method (the method of unitary transformations), which differs from the existing ones, for studying the stability and the norm of solutions of regular and singularly perturbed initial-value problems for nonautonomous linear and quasilinear systems of ODE with normal and “almost normal” matrices. Our results generalize similar theorems for the corresponding systems with constant matrices. This method allows one to avoid rather cumbersome traditional analysis, including the Lyapunov function method. For special classes of singularly perturbed problems, the method provides estimates for the norms of solutions in the presence of exponential or power boundary layers; these observations enrich the collection of known results in this field.     

Reconstructed Discontinuous Approximation to Stokes Equation in a Sequential Least Squares Formulation

We propose a new least squares finite element method to solve the Stokes problem with two sequential steps. The approximation spaces are constructed by the patch reconstruction with one unknown per element. For the first step, we reconstruct an approximation space consisting of piecewise curl-free polynomials with zero trace. By this space, we minimize a least squares functional to obtain the numerical approximations to the gradient of the velocity and the pressure. In the second step, we minimize another least squares functional to give the solution to the velocity in the reconstructed piecewise divergence-free space. We derive error estimates for all unknowns under both $L^2$ norms and energy norms. Numerical results in two dimensions and three dimensions verify the convergence rates and demonstrate the great flexibility of our method.     

Characterization of the minimal and maximal operator ideals associated to the tensor norm defined by a sequence space

We characterize the minimal and maximal operator ideals associated, in the sense of Defant and Floret, to a wide class of tensor norms derived from a Banach sequence space. Our results are extensions of classical ones about tensor norms of Saphar [Studia Math. 38 (1972) 71-100] and show the key role played by the structure of finite-dimensional subspaces in this kind of problems.     

Analysis of the convergence of the minimal and the orthogonal residual methods

We consider two Krylov subspace methods for solving linear systems, which are the minimal residual method and the orthogonal residual method. These two methods are studied without referring to any particular implementations. By using the Petrov–Galerkin condition, we describe the residual norms of these two methods in terms of Krylov vectors, and the relationship between there two norms. We define the Ritz singular values, and prove that the convergence of these two methods is governed by the convergence of the Ritz singular values. AMS subject classification 65F10     

A Generalization of the Arnold-Wendland Lemma to a Modified Collocation Method for Boundary Integral Equations in ℝ3

We prove quasioptimal and optimal order estimates in various Sobolev norms for the approximation of linear strongly elliptic periodic pseudodifferential equations in two independent variables by a modified method of nodal collocation by odd degree polynomial splines. In the one-dimensional case, our method coincides with the method of nodal collocation when odd degree polynomial splines are employed for the trial functions. The convergence analysis is based on an equivalence which we establish between our method and a nonstandard Galerkin method for an operator closely related to the given operator. This equivalence is realized through a crucial intermediate result (which we now term the Arnold-Wendland lemma) to connect the solution of central finite difference equations and that of certain nonstandard Galerkin equations. The results of this paper are genuine two-dimensional generalizations of the results obtained by ARNOLD and WENDLAND in [2] for the one-dimensional equations.     

The contraction number of a multigrid method with mesh ratio 2 for solving Poisson's equation

The convergence rate of a multigrid method for the solution of Poisson's equation on a uniform grid is estimated. In contrast to recent results of Braess, no intermediate grids are used. Refined estimates of Gauss-Seidel relaxation by weak norms, a strengthened Cauchy inequality, and a duality argument are central. We obtain 0.273 as an upper bound for the contraction number of the two-level procedure. The results hold for arbitrary convex polygonal regions and are independent of the smoothness of the solution.     

Justification of the coupled-mode approximation for a nonlinear elliptic problem with a periodic potential

Coupled-mode systems are used in physical literature to simplify the nonlinear Maxwell and Gross–Pitaevskii equations with a small periodic potential and to approximate localized solutions called gap solitons by analytical expressions involving hyperbolic functions. We justify the use of the 1D stationary coupled-mode system for a relevant elliptic problem by employing the method of Lyapunov–Schmidt reductions in Fourier space. In particular, existence of periodic/anti-periodic and decaying solutions is proved and the error terms are controlled in suitable norms. The use of multi-dimensional stationary coupled-mode systems is justified for analysis of bifurcations of periodic/anti-periodic solutions in a small multi-dimensional periodic potential.     

On an iteration method for a nonlinear differential-operator equation

We study an iteration method for a first-order differential-operator equation with a nonlinear operator in a separable Hilbert space. The convergence of the iterative process is proved in the strong norms. Convergence estimates are derived. We present an application of the suggested method to the solution of a model initial-boundary value problem for a fourth-order parabolic equation.     

On the affine deformation of a Minkowski norm

We prove that the affine deformation of a Minkowski norm is a Minkowski norm. Some important classical norms are derived by using the affine deformation recipe. Applications are done for some special Finsler manifolds.     

On a characteristic property of the Lorentz space in the class of symmetric spaces

We establish a criterion for the equivalence of norms in a symmetric space and in the Lorentz space, in particular, a criterion for the equivalence of norms in an Orlicz space and in the Lorentz space.Translated from Matematicheskie Zametki, Vol. 7, No. 5, pp. 627–633, May, 1970.     

A boundary element Galerkin method for a hypersingular integral equation on open surfaces

A hypersingular boundary integral equation of the first kind on an open surface piece Γ is solved approximately using the Galerkin method. As boundary elements on rectangles we use continuous, piecewise bilinear functions which vanish on the boundary of Γ. We show how to compensate for the effect of the edge and corner singularities of the true solution of the integral equation by using an appropriately graded mesh and obtain the same convergence rate as for the case of a smooth solution. We also derive asymptotic error estimates in lower-order Sobolev norms via the Aubin–Nitsche trick. Numerical experiments for the Galerkin method with piecewise linear functions on triangles demonstrate the effect of graded meshes and show experimental rates of convergence which underline the theoretical results.     

On the stability of the collocation method for the double layer operator on polyhedral domains in R3

We develop a new method to give estimates for the double layer operator on cones in R³. Here we use weighted norms which are equivalent to the usual L^∞‐norm. This result includes the weighted norms which were constructed by Wendland and Kral for the case of rectangular cones. If all vertices in a polyhedral domain (resp. their corresponding cones) allow the construction of a weighted norm, such that the double layer operator has norm smaller than one half, we can prove the stability of the collocation method with piecewise constant trial functions.     

Approximation by solutions of elliptic equations in semilocal spaces

In the present work, we investigate the approximability of solutions of elliptic partial differential equations in a bounded domain Ω by solutions of the same equations in a larger domain. We construct an abstract framework which allows us to deal with such density questions, simultaneously for various norms. More specifically, we study approximations with respect to the norms of semilocal Banach spaces of distributions. These spaces are required to satisfy certain postulates. We establish density results for elliptic operators with constant coefficients which unify and extend previous results. In our density results Ω may possess holes and it is required to satisfy the segment condition. We observe that analogous density results do not hold in spaces where the infinitely smooth functions are not dense. Finally, we provide applications related to the method of fundamental solutions.     

Input-output behaviour of a stochastic Galerkin method for a low pass filter

We apply the stochastic Galerkin method to a linear dynamical system, which includes random variables to quantify uncertainties in physical parameters. The input-output behaviour of the stochastic Galerkin system is described by a transfer function in the frequency domain. The importance of each output component can be estimated by Hardy norms. We investigate a Hardy norm in the case of a linear dynamical system modelling the electric circuit of a low pass filter. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Cubic-regularization counterpart of a variable-norm trust-region method for unconstrained minimization

In a recent paper, we introduced a trust-region method with variable norms for unconstrained minimization, we proved standard asymptotic convergence results, and we discussed the impact of this method in global optimization. Here we will show that, with a simple modification with respect to the sufficient descent condition and replacing the trust-region approach with a suitable cubic regularization, the complexity of this method for finding approximate first-order stationary points is \(O(\varepsilon ^{-3/2})\). We also prove a complexity result with respect to second-order stationarity. Some numerical experiments are also presented to illustrate the effect of the modification on practical performance.     

A mixed method for axisymmetric div-curl systems

We present a mixed method for a three-dimensional axisymmetric div-curl system reduced to a two-dimensional computational domain via cylindrical coordinates. We show that when the meridian axisymmetric Maxwell problem is approximated by a mixed method using the lowest order elements (for the vector variable) and linear elements (for the Lagrange multiplier), one obtains optimal error estimates in certain weighted Sobolev norms. The main ingredient of the analysis is a sequence of projectors in the weighted norms satisfying some commutativity properties.

     

Inverted finite elements for div-curl systems in the whole space

We use inverted finite element method (IFEM) for computing three-dimensional vector potentials and for solving div-curl systems in the whole space \(\mathbb {R}^{3}\). IFEM is substantially different from the existing approaches since it is a non truncature method which preserves the unboundness of the domain. After developping the method, we analyze its convergence in term of weighted norms. We then give some three-dimensional numerical results which demonstrate the efficiency and the accuracy of the method and confirm its convergence.     

Convergence of a subgradient method for computing the bound norm of matrices

Let φ and ψ be any norms on $\text{R}$^m and $\text{R}$ⁿ respectively. We study a subgradient method for computing the associated bound norm S_φψ(A) = sup{φ(Ax), ψ(x)?1} (a nonconvex optimization problem). It is proved that homodual method converges when one of the norms φ and ψ is polyhedral.     

Newton iterations without inverting the Derivative

We describe an iteration procedure in a Banach space which is quadratically convergent like Newton's method. In fact, it is a modification of the latter. The inverted derivatives are replaced by so-called contractors which are constructed recursively. Moreover, this method is extended to a scale of Banach spaces. It turns out that the rate of convergence remains quadratic, even if the norms of the contractors are increasing exponentially. A hard implicit function theorem results. In particular, this theorem can be applied to prove existence of quasiperiodic solutions for the Lorenz model of stationary convection.