 In this paper, we present "a posteriori" analysis of the fundamental concepts involved in the modelling of problems of mathematical physics by Partial Differential Equations (PDEs). Our aim is to improve our students' understanding of PDEs when applied to an engineering problem, from a completely qualitative point of view. They should be able to understand the deep meaning of any Laplacian, c...

In this paper, a novel wave equation for electromagnetic waves in a medium having anisotropic permittivity has been derived with the help of Maxwell’s curl equations. The x and y components of the Maxwell’s equations are written with the permittivity ( ) being a 3× 3 symmetric matrix. These equations are solved for Ex , Ey, Hx, Hy in terms of Ez , Hz, and the partial derivatives. The Z componen...

In this paper, we focus on the representation of a divergence-free vector field, defined, on a connected nonsimply connected domain Ω ⊂ R3 with a connected boundary Γ, by its curl and its normal component on the boundary. The considered problem is discretized with H(curl)and H(div)-conforming finite elements. In order to ensure the uniqueness of the vector potential, we propose a spanning tree ...

We study the linearization of three dimensional Regge calculus around Euclidean metric. We provide an explicit formula for the corresponding quadratic form and relate it to the curlt curl operator which appears in the quadratic part of the Einstein-Hilbert action and also in the linear elasticity complex. We insert Regge metrics in a discrete version of this complex, equipped with densily defin...

We present an effective construction of divergence-free wavelets on the square, with suitable boundary conditions. Since 2D divergence-free vector functions are the curl of scalar stream-functions, we simply derive divergence-free multiresolution spaces and wavelets by considering the curl of standard biorthogonal multiresolution analyses (BMRAs) on the square. The key point of the theory is th...

We propose a variational method to simultaneously detect dislocations and grain boundaries in an image of a crystal as well as the local crystal distortion. To this end we extract a distortion field F from the image which is nearly curl-free except at dislocations and grain boundaries. The sparsity of the curl is promoted by an L1-regularization of curlF in the employed energy functional. The s...

LOSS OF CASPASE-2 DEPENDENT APOPTOSIS INDUCES AUTOPHAGY FOLLOWING MITOCHONDRIA OXIDATIVE STRESS IN PRIMARY CULTURES OF YOUNG-ADULT CORTICAL NEURONS Meenakshi Tiwari, Marisa Lopez-Cruzan, William W. Morgan, Brian Herman From Department of Cellular and Structural Biology, University of Texas Health Science Center at San Antonio, 7703 Floyd Curl Drive, San Antonio, TX 78229, USA Running Title: Cas...

The local discontinuous Galerkin method for the numerical approximation of the time-harmonic Maxwell equations in a low-frequency regime is introduced and analyzed. Topologically nontrivial domains and heterogeneous media are considered, containing both conducting and insulating materials. The presented method involves discontinuous Galerkin discretizations of the curl-curl and grad-div operato...

Since their popularization in the late 1970s and early 1980s, multigrid methods have been a central tool in the numerical solution of the linear and nonlinear systems that arise from the discretization of many PDEs. In this paper, we present a local Fourier analysis (LFA, or local mode analysis) framework for analyzing the complementarity between relaxation and coarse-grid correction within mul...

We derive results on equivalence of piecewise polynomial approximations a given function in the Sobolev space $${\varvec{H}}{\mathrm{(curl)}}$$ . namely show that global-best approximation -conforming imposing continuity tangential trace can be bounded above and below by Hilbertian sum respective local from elementwise spaces without any inter-element requirement. In other words, tangential-tra...