Fremlin tensor product behaves well with the unbounded order convergence

Let L be a vector lattice, "(" x_α ") " be a L-valued net, and x∈L . If |x_α-x|∧u→┴o 0 for every u ∈〖 L〗_+ then it is said that the net "(" x_α ")" unbounded order converges to x and is denoted by □(x_α □(→┴uo x)) . This definition of unbounded order convergence has been extensively studied on many structures, including vector lattices, local solid vector lattices, normed lattices and lattice normed spaces. It is not possible to apply this type of convergence to general lattices due to the lack of algebraic structure. Therefore, we will use a type of convergence that is considered to be the motivation for this type of convergence, first defined as independent convergence in semi-ordered linear spaces and later called unbounded order convergence. Namely, L is a lattice, x_α is an L -valued net, and x ϵ L . If (x_α∧b )∨a order converges to (x∧b )∨a for every a,b∈L with a≤b, then it is said that "(" x_α ")" individual converges to x or unbounded order converges to x . This definition can be easily applied to general lattices. In this article, this definition will be understood as unbounded order convergence. Also, even if these two convergences are called by the same name, there is no equivalence between them for general lattices, an example of this is mentioned in this article. Let L be a partially ordered set, "(" x_α ")" be an L -valued net and x∈L (x_α) is said to be star convergent to x if every subnet of the net (x_α ) has a subnet that is order convergent to x and denoted by x_α □(→┴s x). In this paper, a new type of convergence on lattices is defined by combining unbounded order convergence (individual convergence) and star convergence. Let L be a lattice, (x_α ) a net and x∈L (x_α) is said to be unbounded star convergent to x if for every subnet (x_β) of (x_α), there exists a subnet (x_ζ) of (x_β) such that (x_ζ∧b)∨ □(a→┴o ) (x∧b)∨a for every a,b∈L with a≤b and it is denoted by x_α □(→┴us x). The differences between the new type of convergence, called unbounded star convergence, and order convergence, star convergence are demonstrated with counterexamples. The meaningfulness of the unbounded star convergence type is analyzed with these counterexamples and the implications presented. In addition, basic questions about unbounded star convergence of a given net on lattices such as convergence of a fixed net, uniqueness of the limit, convergence of the subnet of a convergent net are answered.

For vector lattices E and F, where F is Dedekind complete and supplied with a locally solid topology, we introduce the corresponding locally solid absolute strong operator topology on the order bounded operators {mathscr{L}}_{mathrm{ob}}(E,F) from E into F. Using this, it follows that {mathscr{L}}_{mathrm{ob}}(E,F) admits a Hausdorff uo-Lebesgue topology whenever F does. For each of order convergence, unbounded order convergence, and—when applicable—convergence in the Hausdorff uo-Lebesgue topology, there are both a uniform and a strong convergence structure on {mathscr{L}}_{mathrm{ob}}(E,F). Of the six conceivable inclusions within these three pairs, only one is generally valid. On the orthomorphisms of a Dedekind complete vector lattice, however, five are generally valid, and the sixth is valid for order bounded nets. The latter condition is redundant in the case of sequences of orthomorphisms, as a consequence of a uniform order boundedness principle for orthomorphisms that we establish. We furthermore show that, in contrast to general order bounded operators, orthomorphisms preserve not only order convergence of nets, but unbounded order convergence and—when applicable—convergence in the Hausdorff uo-Lebesgue topology as well.

Unbounded order convergence in dual spaces

In order to find an extension of Brezis–Lieb’s lemma to the case of nets, we replace the almost everywhere convergence by the unbounded order convergence and introduce the pre-Brezis–Lieb property in normed lattices. Then we identify a wide class of Banach lattices in which the Brezis–Lieb lemma holds true. Among other things, it gives an extension of the Brezis–Lieb lemma for nets in \(L^p\) for \(p\in [1,\infty )\).

Several recent papers investigated unbounded convergences in Banach lattices. The focus of this paper is to apply the results of unbounded convergence to the classical Banach lattice theory from a new perspective. Combining all unbounded convergences, including unbounded order (norm, absolute weak, absolute weak*) convergence, we characterize L-weakly compact sets, L-weakly compact operators and M-weakly compact operators on Banach lattices. For applications, we introduce so-called statistical-unbounded convergence and use these convergences to describe KB-spaces and reflexive Banach lattices.

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In this paper, we provide the first a posteriori error analysis of the discontinuous Galerkin (DG) method for solving the two-dimensional linear hyperbolic conservation laws on Cartesian grids. The key ingredients in our error analysis are the recent optimal superconvergence results proved in Cao et al. (SIAM J Numer Anal 53:1651---1671, 2015). We first prove that the DG solution converges in the $$L^2$$L2-norm to a Radau interpolating polynomial under mesh refinement. The order of convergence is proved to be $$p+2$$p+2, when tensor product polynomials of degree at most p are used. Then we show that the actual error can be divided into a significant part and a less significant part. The significant part of the DG error is spanned by two $$(p+1)$$(p+1)-degree right Radau polynomials in the x and y directions. The less significant part converges to zero at $${\mathcal {O}}\left( h^{p+2}\right) $$Ohp+2. These results are used to construct simple, efficient and asymptotically exact a posteriori error estimates. Superconvergence towards the right Radau interpolating polynomial is used to prove that, for smooth solutions, our a posteriori DG error estimates converge at a fixed time to the true spatial errors in the $$L^2$$L2-norm at $${\mathcal {O}}\left( h^{p+2}\right) $$Ohp+2 rate. Finally, we prove that the global effectivity indices in the $$L^2$$L2-norm converge to unity at $${\mathcal {O}}(h)$$O(h) rate. Our proofs are valid for arbitrary regular Cartesian meshes using tensor product polynomials of degree at most p and for both the periodic and Dirichlet boundary conditions. Several numerical experiments are performed to validate the theoretical results.

Finite element method for elliptic problems with edge singularities

Discretely nonlinearly stable weight-adjusted flux reconstruction high-order method for compressible flows on curvilinear grids

A Crank-Nicolson WG-FEM for unsteady 2D convection-diffusion equation with nonlinear reaction term on layer adapted mesh

We design and analyze several Finite Element Methods (FEMs) applied to the Caffarelli-Silvestre extension that localizes the fractional powers of symmetric, coercive, linear elliptic operators in bounded domains with Dirichlet boundary conditions. We consider open, bounded, polytopal but not necessarily convex domains $\Omega \subset \mathbb{R}^d$ with $d=1,2$. For the solution to the extension problem, we establish analytic regularity with respect to the extended variable $y\in (0,\infty)$. We prove that the solution belongs to countably normed, power-exponentially weighted Bochner spaces of analytic functions with respect to $y$, taking values in corner-weighted Kondat'ev type Sobolev spaces in $\Omega$. In $\Omega\subset \mathbb{R}^d$, we discretize with continuous, piecewise linear, Lagrangian FEM ($P_1$-FEM) with mesh refinement near corners, and prove that first order convergence rate is attained for compatible data $f\in \mathbb{H}^{1-s}(\Omega)$. We also prove that tensorization of a $P_1$-FEM in $\Omega$ with a suitable $hp$-FEM in the extended variable achieves log-linear complexity with respect to $\mathcal{N}_\Omega$, the number of degrees of freedom in the domain $\Omega$. In addition, we propose a novel, sparse tensor product FEM based on a multilevel $P_1$-FEM in $\Omega$ and on a $P_1$-FEM on radical-geometric meshes in the extended variable. We prove that this approach also achieves log-linear complexity with respect to $\mathcal{N}_\Omega$. Finally, under the stronger assumption that the data is analytic in $\overline{\Omega}$, and without compatibility at $\partial \Omega$, we establish exponential rates of convergence of $hp$-FEM for spectral, fractional diffusion operators. We also report numerical experiments for model problems which confirm the theoretical results. We indicate several extensions and generalizations of the proposed methods.

We present stable mixed finite elements for planar linear elasticity on general quadrilateral meshes. The symmetry of the stress tensor is imposed weakly and so there are three primary variables, the stress tensor, the displacement vector field, and the scalar rotation. We develop and analyze a stable family of methods, indexed by an integer r ? 2 and with rate of convergence in the L2 norm of order r for all the variables. The methods use Raviart---Thomas elements for the stress, piecewise tensor product polynomials for the displacement, and piecewise polynomials for the rotation. We also present a simple first order element, not belonging to this family. It uses the lowest order BDM elements for the stress, and piecewise constants for the displacement and rotation, and achieves first order convergence for all three variables.