We propose and analyze a new hybridizable discontinuous Galerkin (HDG) method for second-order elliptic problems. Our method is obtained by inserting the L-orthogonal projection onto the approximate space for a numerical trace into all facet integrals in the usual HDG formulation. The orders of convergence for all variables are optimal if we use polynomials of degree k + l, k + 1 and k, where k...

in this paper boundedness and compactness of generalized composition oper-ators from logarithmic bloch type spaces to q_k type spaces are investigated.

In this paper, by means of the concept of attractive points of a nonlinear mapping, we prove strong convergence theorem of the Ishikawa iteration for an (α, β)−generalized hybrid mapping in a uniformly convex Banach space, and obtain weak convergence theorem of the Ishikawa iteration for such a mapping in a Hilbert space.

Coarse spaces are instrumental in obtaining scalability for domain decomposition methods. However, it is known that most popular choices of coarse spaces perform rather weakly in presence of heterogeneities in the coefficients in the partial differential equations, especially for systems. Here, we introduce in a variational setting a new coarse space that is robust even when there are such hete...

We develop a fast and accurate solver for the forward problem of diffusion tomography in the case of several spherical inhomogeneities. The approach allows one to take into account multiple scattering of diffuse waves between different inhomogeneities. Theoretical results are illustrated with numerical examples; excellent numerical convergence and efficiency are demonstrated. The method is gene...

‎let $x,y$ be normed spaces with $l(x,y)$ the space of continuous‎ ‎linear operators from $x$ into $y$‎. ‎if ${t_{j}}$ is a sequence in $l(x,y)$,‎ ‎the (bounded) multiplier space for the series $sum t_{j}$ is defined to be‎ [ ‎m^{infty}(sum t_{j})={{x_{j}}in l^{infty}(x):sum_{j=1}^{infty}%‎ ‎t_{j}x_{j}text{ }converges}‎ ‎]‎ ‎and the summing operator $s:m^{infty}(sum t_{j})rightarrow y$ associat...

In this paper we introduce a broad class of nonlinear mappings which contains the class of contractive mappings and the class of generalized hybrid mappings in a Hilbert space. Then we prove an attractive point theorem for such mappings in a Hilbert space. Furthermore, we prove a mean convergence theorem of Baillon’s type without convexity in a Hilbert space. Finally, we prove a weak convergenc...

We propose a generalized proximal point algorithm (PPA), in the generic setting of finding a root of a maximal monotone operator. In addition to the classical PPA, a number of benchmark operator splitting methods in the PDE and optimization literatures can be retrieved by this generalized PPA scheme. We establish the convergence rate of this generalized PPA scheme under different conditions, in...

We propose a multiscale method based on a finite element heterogeneous multiscale method (in space) and the implicit Euler integrator (in time) to solve nonlinear monotone parabolic problems with multiple scales due to spatial heterogeneities varying rapidly at a microscopic scale. The multiscale method approximates the homogenized solution at computational cost independent of the small scale b...

We study “circular net” (discrete orthogonal net) equations for angular data generalized by external spectral parameters. A criterion defining physical regimes of these Hamiltonian equations is the reality of Lagrangian density. There are four distinct regimes for fields and spectral parameters classified by four types of spherical or hyperbolic triangles. Non-zero external spectral parameters ...