a mixed finite element formulation for maxwell s equations in the time domain abstract a galerkin finite element solution technique for the maxwell s equations is discussed this new formulation can be viewed as a generalization of certain staggered grid finite difference schemes to arbitrary meshes it is shown that this technique is simple to implement and is more accurate as well as more cost effective than the standard equal order finite element approach numerical are presented to evaluate the performance of this new element relative to the standard element

the orthogonal decomposition theorems for mimetic finite difference methods accurate discrete analogs of differential operators that satisfy the identities and theorems of vector and tensor calculus provide reliable finite difference methods for approximating the solutions to a wide class of partial differential equations these methods mimic many fundamental properties of the underlying physical problem including conservation laws symmetries in the solution and the nondivergence of particular vector fields i e they are divergence free and should satisfy a discrete version of the orthogonal decomposition theorem this theorem plays a fundamental role in the theory of generalized solutions and in the numerical solution of physical models including the navier stokes equations and in electrodynamics we are deriving mimetic finite difference approximations of the divergence gradient and curl that satisfy discrete analogs of the integral identities satisfied by the differential operators we first define the natural discrete divergence gradient and curl operators based on coordinate invariant definitions such as gauss s theorem for the divergence next we use the formal adjoints of these natural operators to derive compatible divergence gradient and curl operators with complementary domains and ranges of values in this paper we prove that these operators satisfy discrete analogs of the orthogonal decomposition theorem and demonstrate how a discrete vector can be decomposed into two orthogonal vectors in a unique way satisfying a discrete analog of the formula vec a ggrad varphi curl vec b we also present a numerical example to illustrate the numerical procedure and calculate the convergence rate of the method for a spiral vector field

a boundedness criterion for generalized calderon zygmund operators in 1965 a p calderon showed the l2 boundedness of the so called first calderon commutator this is one of the first examples of a non convolution operator associated to a kernel which satisfies certain size and smoothness properties comparable to those of the kernel of the hilbert transform these properties together with the l2 boundedness imply the lp boundedness for all p s in 1 oo many operators in analysis such as certain classes of pseudo differential operators and the cauchy integral operator on a curve are associated with kernels having these properties for these operators one of the major questions is if they are bounded on l2 we are going to give necessary and sufficient conditions for such an operator to be bounded on l2 they are essentially that the images of the function 1 under the actions of the operator and its adjoint both lie in bmo in the case of the aforementioned first commutator this can be checked by an integration by parts in the first section we present some basic notions and state the theorem which is proved in sections 2 and 3 in section 4 we show how to recover some classical results in sections 5 and 6 we construct a functional calculus for small perturbations of a and in section 7 we show a connection between the theory of calderon zygmund operators and kato s conjecture it is a pleasure to express our thanks to r r coifman and y meyer for suggesting many elegant simplifications in our proofs and most of the applications we also wish to thank stephen semmes for several pertinent remarks

analysis of a finite element method for maxwell s equations the use of finite elements to discretize the time dependent maxwell equations on a bounded domain in three dimensional space is analyzed energy norm error estimates are provided when general finite element methods are used to discretize the equations in space in addition it is shown that if some curl conforming elements due to nedelec are used error estimates may also be proved in the l 2 norm

the numerical solution of diffusion problems in strongly heterogeneous non isotropic materials a new second order finite difference algorithm for the numerical solution of diffusion problems in strongly heterogeneous and non isotropic media is constructed on problems with rough coefficients or highly nonuniform grids the new algorithm is superior to all other algorithms we have compared it with for problems with smooth coefficients on smooth grids the method is comparable with other second order methods the new algorithm is formulated for logically rectangular grids and is derived using thesupport operators method a key idea in deriving the method was to replace the usual inner product of vector functions by an inner product weighted by the inverse of the material properties tensor and to use the flux operator defined as the material properties tensor times the gradient rather than the gradient as one of the basic first order operators in the support operators method the discrete analog of the flux operator must also be the negative adjoint of the discrete divergence in an inner product that is a discrete analog of the continuum inner product the resulting method is conservative and the discrete analog of the variable coefficient laplacian is symmetric and negative definite on nonuniform grids in addition on any grid the discrete divergence is zero on constant vectors the null space for the gradient is the constant functions and when the material properties are piecewise constant the discrete flux operator is exact for piecewise linear functions we compare the methods on some of the most difficult examples to be found in the literature

three dimensional modeling of ip effects in time domain electromagnetic data understanding the effects of induced polarization ip effects on time domain electromagnetic data requires the ability to simulate common survey techniques when taking chargeability into account most existing techniques preform this modeling in the frequency domain prior to transforming their results to the time domain even though this technique can allow for chargeable material to be easily incorporated its application for some problems can be computationally limiting we developed a new technique for forward modeling the time domain electromagnetic response of chargeable materials in three dimensions the frequency dependence of ohms law translates to an ordinary differential equation when considered in the time domain the system of ordinary partial differential equations was then discretized using an implicit time stepping algorithm that yielded absolute stability this approach allowed us to operate directly in the time domain and avoid frequency to time domain transformations although this approach can be applied directly to materials exhibiting debye dispersions other cole cole dispersions resulted in fractional derivatives in time to overcome this difficulty pade approximations were used to represent the frequency dependence as a rational series of integer order terms the resulting method was then simplified to generate a reduced time domain model that can be used to forward model the ip decay curves in the absence of any electromagnetic coupling we found numerical examples in which the method produced accurate results the potential application of the method was demonstrated by modeling the full time domain electromagnetic response of a gradient array ip survey and the occurrence of negative transients in airborne time domain electromagnetic data

solving the tensorial 3d acoustic wave equation a mimetic finite difference time domain approach abstractgenerating accurate numerical solutions of the acoustic wave equation awe is a key computational kernel for many seismic imaging and inversion problems although finite difference time domain fdtd approaches for generating full wavefield solutions are well developed for cartesian computational domains several challenges remain when applying fdtd approaches to scenarios arguably best described by more generalized geometry in particular how best to generate accurate and stable fdtd solutions for scenarios involving grids conforming to complex topography or internal surfaces we address these issues by developing a mimetic fdtd mfdtd approach that combines four key components a tensorial 3d awe mimetic finite difference mfd operators fully staggered grids fsgs and mfd robin boundary conditions rbc the tensorial formulation of the 3d awe permits wave propagation to be specified on semi analytically defined coordinate meshes designed to conform to complex domain boundaries mfd o

an analysis of ne de lec s method for the spatial discretization of maxwell s equations abstract in 1980 nedelec developed a family of curl and divergence conforming finite elements in r 3 he proposed the use of these elements to discretize the time dependent maxwell equations noting that the elements have the advantage that the discrete magnetic displacement can be made exactly divergence free in this paper we shall analyze a slight generalization of nedelec s scheme and prove essentially optimal order convergence estimates in a variety of situations we also demonstrate that the nedelec method can be superconvergent at certain special points and we relate the method to yee s finite difference scheme a by product of our analysis will be a convergence proof for yee s method

geometric bvps hardy spaces and the cauchy integral and transform on regions with corners abstract in this paper we give a new perspective on the cauchy integral and transform and hardy spaces for dirac type operators on manifolds with corners of codimension two instead of considering banach or hilbert spaces we use polyhomogeneous functions on a geometrically blown up version of the manifold called the total boundary blow up introduced by mazzeo and melrose r r mazzeo r b melrose analytic surgery and the eta invariant geom funct anal 5 1 1995 14 75 these polyhomogeneous functions are smooth everywhere on the original manifold except at the corners where they have a taylor series with possible log terms in polar coordinates the main application of our analysis is a complete fredholm theory for boundary value problems of dirac operators on manifolds with corners of codimension two

a nondiffusive finite volume scheme for the three dimensional maxwell s equations on unstructured meshes we prove a sufficient cfl like condition for the l2 stability of the second order accurate finite volume scheme proposed by remaki for the time domain solution of maxwell s equations in heterogeneous media with metallic and absorbing boundary conditions we yield a very general sufficient condition valid for any finite volume partition in two and three space dimensions numerical tests show the potential of this original finite volume scheme in one two and three space dimensions for the numerical solution of maxwell s equations in the time domain

approximation of boundary conditions for mimetic finite difference methods abstract the numerical solution of partial differential equations solved with finite difference approximations that mimic the symmetry properties of the continuum differential operators and satisfy discrete versions of the appropriate integral identities are more likely to produce physically faithful results furthermore those properties are often needed when using the energy method to prove convergence and stability of a particular difference approximation unless special care is taken mimetic difference approximations derived for the interior grid points will fail to preserve the symmetries and identities between the gradient curl and divergence operators at the computational boundary in this paper we describe how to incorporate boundary conditions into finite difference methods so the resulting approximations mimic the identities for the differential operators of vector and tensor calculus the approach is valid for a wide class of partial differential equations of mathematical physics and will be described for poisson s equation with dirichlet neumann and robin boundary conditions we prove that the resulting difference approximation is symmetric and positive definite for each of these boundary conditions

squared distance matrix of a tree inverse and inertia abstract let t be a tree with vertices v t 1 n the distance between vertices i j v t denoted d i j is defined to be the length the number of edges of the path from i to j we set d i i 0 i 1 n the squared distance matrix of t is the n n matrix with i j element equal to 0 if i j and d i j 2 if i j it is known that is nonsingular if and only if the tree has at most one vertex of degree 2 we obtain a formula for 1 if it exists when the tree has no vertex of degree 2 the formula is particularly simple and depends on a certain two step laplacian of the tree we determine the inertia of the inverse and the inertia of the edge orientation matrix are also described

on distance matrices and laplacians we consider distance matrices of certain graphs and of points chosen in a rectangular grid formulae for the inverse and the determinant of the distance matrix of a weighted tree are obtained results concerning the inertia and the determinant of the distance matrix of an unweighted unicyclic graph are proved if d is the distance matrix of a tree then we obtain certain results for a perturbation of d 1 as an example it is shown that if l is the laplacian matrix of an arbitrary connected graph then d 1 l 1 is an entrywise positive matrix we consider the distance matrix of a subset of a rectangular grid of points in the plane if we choose m k 1 points not containing a closed path in an m k grid then a formula for the determinant of the distance matrix of such points is obtained

chord arc curves and the beurling transform we study the relation between the geometric properties of a quasicircle gamma and the complex dilatation mu of a quasiconformal mapping that maps the real line onto gamma denoting by s the beurling transform we characterize bishop jones quasicircles in terms of the boundedness of the operator i mu s on a particular weighted l 2 space and chord arc curves in terms of its invertibility as an application we recover the l 2 boundedness of the cauchy integral on chord arc curves

support operator finite difference algorithms for general elliptic problems abstract an algorithm is developed for discretizing boundary value problems given by a general linear elliptic second order partial differential equation with general mixed or robin boundary conditions in general logically rectangular grids the continuum problem can be written as an operator equation where the operator is self adjoint and positive definite the discrete approximations have the same property consequently the matrices for the discrete problem are symmetric and positive definite also the scheme has a nearest neighbor stencil consequently the most powerful linear solvers can be applied in smooth grids the algorithm produces second order accurate solutions it is the generality of the problem general matrix coefficients general boundary conditions general logically rectangular grids that makes finding such an algorithm difficult the algorithm which is a combination of the method of support operators and the mapping method overcomes certain difficulties of the individual methods producing a high quality algorithm for solving general elliptic problems

a boundary difference method for electromagnetic scattering problems with perfect conductors and corners boundary integral equations bie are widely used in applied electromagnetics with the boundary element method bem typically employed for numerical solution the paper extends the recently proposed boundary difference method bdm an alternative to bem to electromagnetic scattering problems with perfect conductors and with corners bdm shares several key features and advantages of bem but avoids the singular kernels inherent in bie convergence rates of the method established via numerical experiments are in agreement with the theoretical predictions

high order mimetic finite difference operators satisfying the extended gauss divergence theorem abstract we present high order mimetic finite difference operators that satisfy the extended gauss theorem these operators have the same order of accuracy in the interior and at the boundary no free parameters and optimal bandwidth they are defined over staggered grids using weighted inner products with a diagonal norm we present several examples to demonstrate that mimetic finite difference schemes using these operators produce excellent results

analyticity of layer potentials and l 2 solvability of boundary value problems for divergence form elliptic equations with complex l infty coefficients we consider divergence form elliptic operators of the form l dv a x nabla defined in r n 1 x t in r n times r n geq 2 where the l infty coefficient matrix a is n 1

convolution singular integrals on lipschitz surfaces we prove the lp boundedness of convolution singular integral op erators on a lipschitz surface g x eo x e r 1 x e rn where g is a lipschitz function which satisfies ilvgli we have embedded r i in the clifford algebra r n with identity eo and are considering convolution with right monogenic functions 0 which satisfy 10 x i 0 so x xo x e r lxol where p a w provided there exists an l function q satisfying 0o r 0 r xern then the related convolution singular integral operator t u x lim o f yei x y n y u y dsy 0 en x u x e o jl is bounded on lp for 1 tronics macquarie university nsw 2109 australia e mail address chun li chun macadam mpce mq edu au e mail address alan mcintosh alan macadam mpce mq edu au stephen semmes department of mathematics rice university houston texas 77251 e mail address semmes rice edu this content downloaded from 157 55 39 143 on wed 15 jun 2016 04 40 33 utc all use subject to http about jstor org terms

the tb theorem on non homogeneous spaces 0 introduction main objects and results 3 0 1 main results 4 0 2 an application of t1 heorem electric intensity capacity 7 0 3 how to interpret calderon zygmund operator t 9 0 3 1 bilinear form is defined on lipschitz functions 10 0 3 2 bilinear form is defined for smooth functions 11 0 3 3 apriori boundedness 12 0 4 plan of the paper 15

a new mimetic scheme for the acoustic wave equation a new mimetic finite difference scheme for solving the acoustic wave equation is presented it combines a novel second order tensor mimetic discretizations in space and a leapfrog approximation in time to produce an explicit multidimensional scheme convergence analysis of the new scheme on a staggered grid shows that it can take larger time steps than standard finite difference schemes based on ghost points formulation a set of numerical test problems gives evidence of the versatility of the new mimetic scheme for handling general boundary conditions

sharp commutator estimates via harmonic extensions abstract we give an alternative proof of several sharp commutator estimates involving riesz transforms riesz potentials and fractional laplacians our methods only involve harmonic extensions to the upper half space integration by parts and trace space characterizations the commutators we investigate are jacobians more generally coifman rochberg weiss commutators chanillo s commutator with the riesz potential coifman mcintosh meyer kato ponce vega type commutators the fractional leibniz rule and the da lio riviere three term commutators we also give a new limiting l 1 estimate for a double commutator of coifman rochberg weiss type and several intermediate estimates the beauty of our method is that all those commutator estimates which are originally proven by various specific methods or by general para product arguments can be obtained purely from integration by parts and trace theorems another interesting feature is that in all these cases the cancellation effect responsible for the commutator estimate simply follows from the product rule for classical derivatives and can be traced in a precise way

unbounded holomorphic fourier multipliers on starlike lipschitz surfaces and applications to sobolev spaces abstract by a generalization of fueter s result we establish the correspondence between the convolution operators and the fourier multipliers on starlike lipschitz surfaces as applications we obtain the sobolev boundedness of the fourier multipliers and prove the equivalence between two classes of hardy sobolev spaces on starlike lipschitz surfaces

wavelets and the angle between past and future the main result the riesz projectionp or equivalently hilbert transformt is bounded in the weighted spacel2 w wherewis a matrix valued weight if and only ifsupi 1 i iw 1 2 1 i iw 1 1 2 wheresupremumis taken over all intervalsi motivation for this problem comes from stationary processes riesz projection is bounded means the angle between past and future of a stationary process with spectral measurewis nonzero in the scalar case the result is the well known hunt muckenhoupt wheeden theorem the main step in our proof is to show that a vector haar system forms an unconditional basis inl2 w as a byproduct of our approach we get some new results about bases of wavelets in weighted spaces in both scalar and vector valued cases

paraproducts and the exponential square class we prove that if a locally integrable f has a pointwise bounded dyadic square function where the square function is defined with respect to a so called accretive weight b t hen f is locally exponentially square integrable we generalize the result to d dimensions by means of canonical haar functions for l 2 r d 2002 elsevier science usa all rights reserved

fast simulation of 3d electromagnetic problems using potentials we consider solving three dimensional electromagnetic problems in parameter regimes where the quasi static approximation applies the permeability is constant the conductivity may vary significantly and the range of frequencies is moderate the difficulties encountered include handling solution discontinuities across interfaces and accelerating convergence of traditional iterative methods for the solution of the linear systems of algebraic equations that arise when discretizing maxwell s equations in the frequency domain we use a potential current formulation a with a coulomb gauge the potentials a and decompose the electric field e into components in the active and null spaces of the operator we develop a finite volume discretization on a staggered grid that naturally employs harmonic averages for the conductivity at cell faces after discretization we eliminate the current and the resulting large sparse linear system of equations has a block structure that is diagonally dominant allowing an efficient solution with preconditioned krylov space methods a particularly efficient algorithm results from the combination of bicgstab and an incomplete lu decomposition we demonstrate the efficacy of our method in several numerical experiments

cauchy integrals on lipschitz surfaces in octonionic space with the newly developed octonion analytic function theory we confirm the octonionic analogue of the calderon s conjecture as application we obtain the plemelj formula in octonionic space

a decomposition theorem for singular integral operators on spaces of homogeneous type let x d be a space of homogeneous type under the assumption x 0 for all x x we prove a decomposition theorem for singular integral operators on x d isotropic haar expansion gives a representation of the integral operator as a series of simple shifts and rearrangements plus two paraproducts this yields a umd valued t 1 theorem on spaces of homogeneous type

complementary geometric formulations for electrostatics summary the simultaneous use of a pair of complementary discrete formulations for electrostatic boundary value problems bvps allows to accurately compute electromagnetic quantities such as capacitance or electrostatic force with a minimum computational effort in fact the two formulations provide the upper and lower bounds for these quantities and their averages result quite close to the exact solution even for extremely coarse meshes despite the potential benefit to the many three dimensional large scale applications taking advantage of this feature is not exploited in practice due to theoretical difficulties in the potential design the aim of this paper is to fill this gap by rigorously introducing a pair of three dimensional complementary geometric formulations to solve electrostatic bvps on domains covered by conformal polyhedral meshes in particular an original formulation based on a vector potential is introduced by using cohomology theory with integer coefficients it is shown how the so called thick links are needed which are representatives of the second cohomology group generators of the dielectric region two easy to implement graph theoretic algorithms to automatically find such a basis with optimal computational complexity are described some benchmark problems are presented to show how the simultaneous use of both formulations yields to a sensible computational advantage therefore solvers based on complementary formulations should be embedded in the next generation of electromagnetic computer aided engineering cae softwares copyright 2010 john wiley sons ltd

a global tb theorem for compactness and boundedness of calderon zygmund operators abstract we prove a tb theorem that characterizes all calderon zygmund operators that extend compactly on l p r n 1 p the result whose proof does not require the property of accretivity can be used to prove compactness of the double layer potential operator on a wide class of domains the study also provides conditions for boundedness of singular integral operators by means of non accretive testing functions

natural discretizations for the divergence gradient and curl on logically rectangular grids abstract this is the first in series of papers creating a discrete analog of vector analysis on logically rectangular nonorthogonal nonsmooth grids we introduce notations for 2 d logically rectangular grids describe both cell valued and nodal discretizations for scalar functions and construct the natural discretizations of vector fields using the vector components normal and tangential to the cell boundaries we then define natural discrete analogs of the divergence gradient and curl operators based on coordinate invariant definitions and interpret these formulas in terms of curvilinear coordinates such as length of elements of coordinate lines areas of elements of coordinate surfaces and elementary volumes we introduce the discrete volume integral of scalar functions the discrete surface integral and a discrete analog of the line integral and prove discrete versions of the main theorems relating these objects these theorems include the following the discrete analog of relationship div a 0 if and only if a curl b curl a 0 if and only if a grad if a grad then the line integral does not depend on path and if the line integral of a vector function is equal to zero for any closed path then this vector is the gradient of a scalar function last we define the discrete operators div grad and curl in terms of primitive differencing operators based on forward and backward differences and primitive metric operators related to multiplications of discrete functions by length of edges areas of surfaces and volumes of 3 d cells these formulations elucidate the structure of the discrete operators and are useful when investigating the relationships between operators and their adjoints

an octree multigrid method for quasi static maxwell s equations with highly discontinuous coefficients in this paper we develop an octree discretization for maxwell s equations in the quasi static regime we then use this discretization in order to develop a multigrid method for maxwell s equations with highly discontinuous coefficients we test our algorithms and compare it to other multilevel algorithms

regular article finite difference schemes for u t x α δg δu that inherit energy conservation or dissipation property we propose a new procedure for designing by rote finite difference schemes that inherit energy conservation or dissipation property from nonlinear partial differential equations such as the korteweg de vries kdv equation and the cahn hilliard equation the most important feature of our procedure is a rigorous discretization of variational derivatives using summation by parts which implies that the inherited properties are satisfied exactly since the inherited properties are kept even if the time mesh size changes in the time evolution process we can use some appropriate time mesh adaptive methods to obtain numerical solutions through the derived schemes because of these properties the derived schemes are expected to be numerically stable and yield solutions converging to pde solutions and sufficiently flexible to treat the inheritance of the energy conservation and dissipation properties are verified numerically for the kdv equation and the cahn hilliard equation

on boundary properties of solutions of complex vector fields abstract this work presents results on the boundary properties of solutions of a complex planar smooth vector field l classical results in the h p theory of holomorphic functions of one variable are extended to the solutions of a class of nonelliptic complex vector fields

a class of fourier multipliers on starlike lipschitz surfaces abstract in this paper we consider a class of fourier multipliers whose symbols are controlled by a polynomial on starlike lipschitz surfaces and get the l 2 boundedness of these operators on sobolev spaces and their endpoint estimates

convergence analysis of a covolume scheme for maxwell s equations in three dimensions this paper contains error estimates for covolume discretizations of maxwell s equations in three space dimensions several estimates are proved first an estimate for a semi discrete scheme is given second the estimate is extended to cover the classical interlaced time marching technique third some of our unstructured mesh results are specialized to rectangular meshes both uniform and nonuniform by means of some additional analysis it is shown that the spatial convergence rate is one order higher than for the unstructured case

covolume solutions of three dimensional div curl equations the voronoi delaunay mesh systems provide a generalization of the classical rectangular staggered meshes to unstructured meshes in this work it is shown how such covolume discretizations may be applied to div curl systems in three dimensions error estimates are proved and confirmed by a numerical illustration

fourier analysis on starlike lipschitz surfaces a theory of a class of singular integrals on starlike lipschitz surfaces in rn is established the class of singular integrals forms an operator algebra identical to the class of bounded holomorphic fourier multipliers as well as to the cauchy dunford bounded holomorphic functional calculus of the spherical dirac operator the study proposes a new method inducing clifford holomorphic functions from holomorphic functions of one complex variable by means of which problems on the sphere are reduced to those on the unit circle

a framework for the upscaling of the electrical conductivity in the quasi static maxwells equations electromagnetic simulations of complex geologic settings are computationally expensive one reason for this is the fact that a fine mesh is required to accurately discretize the electrical conductivity model of a given setting this conductivity model may vary over several orders of magnitude and these variations can occur over a large range of length scales using a very fine mesh for the discretization of this setting leads to the necessity to solve a large system of equations that is often difficult to deal with to keep the simulations computationally tractable coarse meshes are often employed for the discretization of the model such coarse meshes typically fail to capture the fine scale variations in the conductivity model resulting in inaccuracies in the predicted data in this work we introduce a framework for constructing a coarse mesh or upscaled conductivity model based on a prescribed fine mesh model rather than using analytical expressions we opt to pose upscaling as a parameter estimation problem by solving an optimization problem we obtain a coarse mesh conductivity model the optimization criterion can be tailored to the survey setting in order to produce coarse models that accurately reproduce the predicted data generated on the fine mesh this allows us to upscale arbitrary conductivity structures as well as to better understand the meaning of the upscaled quantity we use 1d and 3d examples to demonstrate that the proposed framework is able to emulate the behavior of the heterogeneity in the fine mesh conductivity model and to produce an accurate description of the desired predicted data obtained by using a coarse mesh in the simulation process

functional calculus of dirac operators and complex perturbations of neumann and dirichlet problems we prove that the neumann dirichlet and regularity problems for divergence form elliptic equations in the half space are well posed in l 2 for small complex l infty perturbations of a coefficient matrix which is either real symmetric of block form or constant all matrices are assumed to be independent of the transversal coordinate we solve the neumann dirichlet and regularity problems through a new boundary operator method which makes use of operators in the functional calculus of an underlaying first order dirac type operator we establish quadratic estimates for this dirac operator which implies that the associated hardy projection operators are bounded and depend continuously on the coefficient matrix we also prove that certain transmission problems for k forms are well posed for small perturbations of block matrices

linear bound for the dyadic paraproduct on weighted lebesgue space l2 w the dyadic paraproduct is bounded in weighted lebesgue spaces lp w if and only if the weight w belongs to the muckenhoupt class apd however the sharp bounds on the norm of the dyadic paraproduct are not known even in the simplest l2 w case in this paper we prove that the bound on the norm of the dyadic paraproduct in the weighted lebesgue space l2 w depends linearly on the a2d characteristic of the weight w using bellman function techniques and extrapolate this result to the lp w case

a second order discretization of maxwell s equations in the quasi static regime on octree grids in this study we consider adaptive mesh refinement for the solution of maxwell s equations in the quasi static or diffusion regime we propose a new finite volume octree discretization for the problem and show how to construct second order stencils on yee grids extending the known first order discretization stencils we then develop an effective preconditioner to the problem we show that our preconditioner performs well for discontinuous conductivities as well as for a wide range of frequencies

a q analogue of the distance matrix of a tree we consider a q analogue of the distance matrix called the q distance matrix of an unweighted tree and give formulae for the inverse and the determinant which generalize the existing formulae for the distance matrix we obtain the smith normal form of the q distance matrix of a tree the relationship of the q distance matrix with the laplacian matrix leads to q analogue of the laplacian matrix of a tree some of whose properties are also given we study another matrix related to the distance matrix the exponential distance matrix and show its relationship with the q laplacian and the q distance matrix a formula for the determinant of the q distance matrix of a weighted tree is also given

efficient time integration of maxwell s equations with generalized finite differences we consider the computationally efficient time integration of maxwell s equations using discrete exterior calculus dec as the computational framework with the theory of dec we associate the degrees of freedom of the electric and magnetic fields with primal and dual mesh structures respectively we concentrate on mesh constructions that imitate the geometry of the close packing in crystal lattices that is typical of elemental metals and intermetallic compounds this class of computational grids has not been used previously in electromagnetics for the simulation of wave propagation driven by time harmonic source terms we provide an optimized hodge operator and a novel time discretization scheme with nonuniform time step size the numerical experiments show a significant improvement in accuracy and a decrease in computing time compared to simulations with well known variants of the finite difference time domain method

mimetic discretizations for maxwell s equations we have constructed reliable finite difference methods for approximating the solution to maxwell s equations using accurate discrete analogs of differential operators that satisfy the identities and theorems of vector and tensor calculus in discrete form the numerical approximation does not have spurious modes and mimics many fundamental properties of the underlying physical problem including conservation laws symmetries in the solution and the nondivergence of particular vector fields numerical examples demonstrate the high quality of the method when the medium is strongly discontinuous and for nonorthogonal nonsmooth computational grids

on the solution of elliptic partial differential equations on regions with corners in this paper we investigate the solution of boundary value problems on polygonal domains for elliptic partial differential equations we observe that when the problems are formulated as the boundary integral equations of classical potential theory the solutions are representable by series of elementary functions in addition to being analytically perspicuous the resulting expressions lend themselves to the construction of accurate and efficient numerical algorithms the results are illustrated by a number of numerical examples

the origin of spurious solutions in computational electromagnetics it is commonly believed that the divergence equations in the maxwell equations are redundant for transient and timeharmonic problems therefore most of the numerical methods in computational electromagnetics solve only two first order curl equations or the second order curl curl equations this misconception is the true origin of spurious modes and inaccurate solutions in computational electromagnetics by studying the div curl system this paper clarifies that the first order maxwell equations are not overdetermined and the divergence equations must always be included to maintain the ellipticity of the system in the space domain to guarantee the uniqueness of the solution and the accuracy of the numerical methods and to eliminate the infinitely degenerate eigenvalue this paper shows that the common derivation and usage of the second order curl curl equations are incorrect and that the solution of helmholtz equations needs the divergence condition to be enforced on an associated part of the boundary the div curl method and the least squares method introduced in this paper provide rigorous derivation of the equivalent second order maxwell equations and their boundary conditions the node based least squares finite element method lsfem is recommended for solving the first order full maxwell equations directly examples of the numerical solutions by lsfem are given to demonstrate that the lsfem is free of spurious solutions