lp gradient estimate for elliptic equations with high contrast conductivities in rn abstract uniform estimate for the solutions of elliptic equations with high contrast conductivities in r n is concerned the space domain consists of a periodic connected sub region and a periodic disconnected matrix block subset the elliptic equations have fast diffusion in the connected sub region and slow diffusion in the disconnected subset suppose 0 1 is the diameter of each matrix block and 2 0 1 is the conductivity ratio of the disconnected matrix block subset to the connected sub region it is proved that the w 1 p norm of the elliptic solutions in the connected sub region is bounded uniformly in when the l p norm of the elliptic solutions in the whole space is bounded uniformly in the w 1 p norm of the elliptic solutions in perforated domains is bounded uniformly in however the l p norm of the second order derivatives of the solutions in the connected sub region may not be bounded uniformly in

green s matrices and boundary estimates in parabolic homogenization abstract for a family of second order parabolic systems in divergence form with rapidly oscillating periodic and time dependent coefficients we establish the pointwise estimates of the green s matrices we investigate uniform boundary lipschitz estimates and boundary holder estimates as well as non tangential maximum function estimates in c 1 or c 1 cylinders consequently we also obtain the agmon miranda maximum principle

non uniform elliptic equations in convex lipschitz domains abstract non uniform elliptic equations in convex lipschitz domains are concerned the non smooth domains consist of a periodic connected high permeability sub region and a periodic disconnected matrix block subset with low permeability let 0 1 denote the size ratio of the matrix blocks to the whole domain and let 2 0 1 denote the permeability ratio of the disconnected matrix block subset to the connected sub region the w 1 p norm for p 1 of the elliptic solutions in the high permeability sub region is shown to be bounded uniformly in however the w 1 p norm of the solutions in the low permeability subset may not be bounded uniformly in roughly speaking if the sources in the low permeability subset are small enough the solutions in that subset are bounded uniformly in otherwise the solutions cannot be bounded uniformly in relations between the sources and the variation of the solutions in the low permeability subset are also presented in this work

a new approximate method for second order elliptic problems with rapidly oscillating coefficients based on the method of multiscale asymptotic expansions abstract in this paper we consider second order elliptic problems with rapidly oscillating coefficients on basis of o a oleinik a s shamaev g a yosifian mathematical problems in elasticity and homogenization north holland amsterdam 1992 wen ming he jun zhi cui a pointwise estimate on the 1 order approximation of g x 0 e ima j appl math 70 2005 241 269 we propose a new approximate method to solve these problems of course we present its error estimate

dimension free quasiconformal distortion in n space most distortion theorems for k quasiconformal mappings in rn n 2 depend on both n and k in an essential way with bounds that become infinite as n tends to oo the present authors obtain dimension free versions of four well known distortion theorems for quasiconformal mappings namely bounds for the linear dilatation the schwarz lemma the e distortion theorem and the r1 quasisymmetry property of these mappings they show that the upper estimates they have obtained in each of these four main results remain bounded as n tends to oo with k fixed the proofs are based on a dimensioncancellation property of the function t h r 1 r t k t 0 k 0 where r t is the capacity of a teichmiiller extremal ring in rn the authors also prove a dimension free distortion theorem for the absolute cross ratio under k quasiconformal mappings of n from which several other distortion theorems follow as special cases

a goal oriented hp adaptive finite element method with electromagnetic applications part i electrostatics we describe the development and application of a finite element fe self adaptive hp goal oriented algorithm for elliptic problems the algorithm delivers without any user interaction a sequence of optimal hp grids this sequence of grids minimizes the error of a prescribed quantity of interest with respect to the problem size the refinement strategy is an extension of a fully automatic energy norm based hp adaptive algorithm we illustrate the efficiency of the method with 2d numerical results among other problems we apply the goal oriented hp adaptive strategy to simulate direct current dc resistivity logging instruments including through casing resistivity tools in a borehole environment and for the assessment of rock formation properties copyright 2005 john wiley sons ltd

goal oriented adaptivity using unconventional error representations for the multidimensional helmholtz equation in goal oriented adaptivity goa the error in the quantity of interest qoi is represented using the error functions of the direct and adjoint problems this error representation is subsequently bounded above by element wise error indicators that are used to drive optimal refinements in this work we propose to replace in the error representation the adjoint problem by an alternative operator the main advantage of the proposed approach is that when judiciously selecting such alternative operator the corresponding upper bound of the error representation becomes sharper leading to a more efficient goa while the method can be applied to a variety of problems we focus here on two and threedimensional 2d and 3d helmholtz problems we show via extensive numerical experimentation that the upper bounds provided by the alternative error representations are sharper than the classical ones and lead to a more robust p adaptive process we also provide guidelines for finding operators delivering sharp error representation upper bounds we further extend the results to a convectiondominated diffusion problem as well as to problems with discontinuous material coefficients finally we consider a sonic logging while drilling lwd problem to illustrate the applicability of the proposed method

pml enhanced with a self adaptive goal oriented hp finite element method simulation of through casing borehole resistivity measurements we describe the application of a perfectly matched layer pml combined with a self adaptive goal oriented hp finite element fe method to the simulation of borehole resistivity measurements the adaptive refinements and fast convergence of the self adaptive hp fe method enhance the performance of the pml thereby enabling the accurate and efficient truncation of the computational domain in open domain problems we apply this method to the simulation of axisymmetric through casing resistivity measurements acquired in a borehole environment that are typically used for the assessment of rock formation properties our numerical results confirm the accuracy and efficiency of our method and provide evidence of highly accurate and reliable simulations of borehole logging measurements in the presence of a conductive steel casing and material contrast of fourteen orders of magnitude in electrical conductivity moreover the combination of adaptivity and pml enables us to significantly reduce the size of the computational domain with substantial savings in computer time and memory

homogenization of elliptic boundary value problems in lipschitz domains in this paper we study the lp boundary value problems for mathcal l u 0 in mathbb r d 1 where mathcal l rm div a nabla is a second order elliptic operator with real and symmetric coefficients assume that a is periodic in xd 1 and satisfies some minimal smoothness condition in the xd 1 variable we show that the lp neumann and regularity problems are uniquely solvable for 1 p 2 we also present a new proof of dahlberg s theorem on the lp dirichlet problem for 2 p dahlberg s original unpublished proof is given in the appendix as the periodic and smoothness conditions are imposed only on the xd 1 variable these results extend directly from mathbb r d 1 to regions above lipschitz graphs consequently by localization techniques we obtain uniform lp estimates for the dirichlet neumann and regularity problems on bounded lipschitz domains for a family of second order elliptic operators arising in the theory of homogenization the results on the neumann and regularity problems are new even for smooth domains

elliptic equations with bmo coefficients in reifenberg domains the inhomogeneous dirichlet problems concerning divergence form elliptic equations are studied optimal regularity requirements on the coefficients and domains for the w1 p theory 1 p are obtained the principal coefficients are supposed to be in the john nirenberg space with small bmo seminorms the domain is a reifenberg domain these conditions for the w1 p theory not only weaken the requirements on the coefficients but also lead to a more general geometric condition on the domains in fact these domains might have fractal dimensions 2004 wiley periodicals inc

uniform regularity estimates in homogenization theory of elliptic system with lower order terms abstract in this paper we extend the uniform regularity estimates obtained by m avellaneda and f lin in 3 6 to the more general second order elliptic systems in divergence form l e e 0 with rapidly oscillating periodic coefficients we establish not only sharp w 1 p estimates holder estimates lipschitz estimates and non tangential maximal function estimates for the dirichlet problem on a bounded c 1 domain but also a sharp o e convergence rate in h 0 1 by virtue of the dirichlet correctors moreover we define the green s matrix associated with l e and obtain its decay estimates we remark that the well known compactness methods are not employed here instead we construct the transformations 1 11 to make full use of the results in 3 6

simulation of wireline sonic logging measurements acquired with borehole eccentered tools using a high order adaptive finite element method the paper introduces a high order adaptive finite element method for simulation of sonic measurements acquired with borehole eccentered logging instruments the resulting frequency domain based algorithm combines a fourier series expansion in one spatial dimension with a two dimensional high order adaptive finite element method fem and incorporates a perfectly matched layer pml for truncation of the computational domain the simulation method was verified for various model problems including a comparison to a semi analytical solution developed specifically for this purpose numerical results indicate that for a wireline sonic tool operating in a fast formation the main propagation modes are insensitive to the distance from the center of the tool to the center of the borehole eccentricity distance however new flexural modes arise with an increase in eccentricity distance in soft formations we identify a new dipole tool mode which arises as a result of tool eccentricity

integration of hp adaptivity and a two grid solver for elliptic problems we present implementation details and analyze convergence of a two grid solver forming the core of the fully automatic hp adaptive strategy for elliptic problems the solver delivers a solution for a fine grid obtained from an arbitrary coarse hp grid by a global hp refinement the classical v cycle algorithm combines an overlapping block jacobi smoother with optimal relaxation and a direct solve on the coarse grid a simple theoretical analysis is illustrated with extensive numerical experimentation

uniform w 1 p estimates for systems of linear elasticity in a periodic medium abstract let l e be a family of elliptic systems of linear elasticity with rapidly oscillating periodic coefficients we obtain the uniform w 1 p estimate u e p c f p in a lipschitz domain in r n for solutions to the dirichlet problem l e u e div f in and u e 0 on where 1 p 1 2 1 2 n and c 0 are constants independent of e 0 the ranges are sharp for n 2 or 3

layer potentials and boundary value problems for elliptic systems in lipschitz domains various layer potential operators are constructed for general elliptic systems of partial differential equations with constant coefficients these operators are used to study the boundary value problems for these systems for a bounded lipschitz domain in rn n 3 it is shown that if the coefficients of the system satisfy the legendre hadamard condition and a symmetry condition then there exists a unique solution u of the system that solves the dirichlet problem for given data g in lp for p in a neighborhood of 2 also the oblique derivative problem for the system is studied furthermore some regularity results for the dirichlet problem are obtained

notes on exterior lp boundary value problems for nonhomogeneous elliptic operators abstract in terms of non smooth domains and non homogeneous elliptic operators this paper is devoted to study non tangential maximal function estimates for exterior l p regular problems we do not repeat the layer potential arguments which is a possible way instead appealing to a radial maximal function we can employ the results on homogeneous operators to make the proof much easier nevertheless this method relies on an exterior w 1 p estimate which is established in the course of the proof

fully automatic hp adaptivity in three dimensions the paper presents a three dimensional implementation of the fully automatic hp adaptive mesh refinement strategy proposed in l demkowicz w rachowicz ph devloo a fully automatic hp adaptivity j sci comput 17 1 3 2002 127 155 the methodology is based on a minimization of the projection based interpolation error with a fixed number of degrees of freedom a coarse grid is first uniformly refined in both h and p to yield a corresponding fine grid a two grid solver is used to solve the problem on the fine grid or iterate towards the fine grid solution the fine grid solution possibly partially converged serves then as a reference solution to be interpolated on the original coarse mesh and the next optimal coarse grid to be determined the difference between the coarse and fine grid solutions provides an excellent error estimate for the coarse grid error the procedure is repeated with the coarse and fine meshes generated during the adaptive process simultaneously until a prescribed error tolerance for the coarse grid error is met the solution obtained on the last fine grid is considered to be the final product delivered by the adaptive procedure

mathematical analysis and finite element simulation of a magnetized ferrite model microwave scattering by obstacles made of ferrite materials has been studied for a long time because of its important applications in radar satellite optical imaging system signal processing atmospheric science and even cloaking device design in this paper we present for the first time the rigorous mathematical analysis of a magnetized ferrite model widely used by engineers the well posedness of the model is established first then we develop a finite element time domain method for solving this model numerical stability similar to the continuous stability is proved for our scheme and the optimal error estimate in the l 2 norm is obtained numerical results are consistent with our theoretical analysis and show how wave propagates in the ferrite material

extensions of goal oriented error estimation methods to simulations of highly nonlinear response of shock loaded elastomer reinforced structures this paper describes extensions of goal oriented methods for a posteriori error estimation and control of numerical approximation to a class of highly nonlinear problems in computational solid mechanics an updated lagrangian formulation of the dynamical large deformation response of structures composed of strain rate sensitive elastomers and elastoplastic materials is developed to apply the theory of goal oriented error estimation a backward in time dual formulation of these problems is derived and residual error estimators for meaningful quantities of interest are established the target problem class is that of axisymmetric deformations of layered elastomer reinforced shells of revolution subjected to shock loading extensive numerical results on solutions of representative problems are given it is shown that extensions of the theory of goal oriented error estimation can be developed and applied effectively to a class of highly nonlinear multi physics problems in solid and structural mechanics

calderon zygmund estimates for stochastic homogenization abstract we prove quenched l p type estimates for the gradient of a solution of a quasilinear elliptic equation with random coefficients

goal oriented error estimation and adaptivity for the finite element method abstract in this paper we study a new approach in a posteriori error estimation in which the numerical error of finite element approximations is estimated in terms of quantities of interest rather than the classical energy norm these so called quantities of interest are characterized by linear functionals on the space of functions to where the solution belongs we present here the theory with respect to a class of elliptic boundary value problems and in particular show how to obtain accurate estimates as well as upper and lower bounds on the error we also study the new concept of goal oriented adaptivity which embodies mesh adaptation procedures designed to control error in specific quantities numerical experiments confirm that such procedures greatly accelerate the attainment of local features of the solution to preset accuracies as compared to traditional adaptive schemes based on energy norm error estimates

gradient estimates for elliptic systems in non smooth domains we obtain the global w 1 p 1 p estimate for the weak solution of an elliptic system with discontinuous coefficients in non smooth domains without using maximal function approach it is assumed that the boundary of a bounded domain is well approximated by hyperplanes at every point and at every scale and that the tensor coefficients belong to bmo space with their bmo semi norms sufficiently small

layer potential methods for elliptic homogenization problems in this paper we use the method of layer potentials to study l 2 boundary value problems in a bounded lipschitz domain for a family of second order elliptic systems with rapidly oscillating periodic coefficients defining l e div a e 1 x under the assumption that a x is elliptic symmetric periodic and holder continuous we establish the solvability of the l 2 dirichlet regularity and neumann problems for l e u e 0 in with optimal estimates uniform in e 0

boundary estimates in homogenization of systems of elasticity on lipschitz domains l p theory abstract the primary purpose of this paper is to investigate a family of elliptic systems of linear elasticity with rapidly oscillating periodic coefficients arising in the theory of homogenization in lipschitz domains as a consequence for d 4 we prove that the l p neumann and l p dirichlet boundary value problems for systems of second order linear elasticity are uniquely solvable for 2 d 1 d 1 p 2 and 2 p 2 d 1 d 3 respectively

uniform regularity estimates in homogenization theory of elliptic systems with lower order terms on the neumann boundary problem abstract in this paper we mainly employed the idea of the previous paper 34 to study the sharp uniform w 1 p estimates with 1 p for more general elliptic systems with the neumann boundary condition on a bounded c 1 domain arising in homogenization theory based on the skills developed by z shen in 27 and by t suslina in 31 32 we also established the l 2 convergence rates on a bounded c 1 1 domain and a lipschitz domain respectively here we found a rough version of the first order correctors see 1 12 which can unify the proof in 27 and 32 it allows us to skip the corresponding convergence results on r d that are the preconditions in 31 32 our results can be regarded as an extension of 23 developed by c kenig f lin z shen as well as of 32 investigated by t suslina

an hp adaptive finite element method for electromagnetics part 1 data structure and constrained approximation abstract this is the first of papers describing an implementation of the hp adaptive mixed finite element fe method for the solution of steady state maxwell s equations proposed in l demkowicz and l vardapetyan modeling of electromagnetic absoption scattering problems using hp adaptive finite elements computer methods in applied mechanics and engineering 152 1998 103 124 the discretization is defined on a hybrid grid consisting of both triangles and quads and allows for both h and p refinements of the mesh the paper focuses on the data structure and constrained approximation issues and provides a number of illustrative examples

on w1 p estimates for elliptic equations in divergence form first we will apply the method to study w 1 p regularity for a nonlinear elliptic operator in divergence form we would like to point out that in the particular case of a linear elliptic equation this method gives an alternative proof to the classical one which uses the general theory of singular integrals the utility of the method that we describe below is that hypotheses a and b are obtained directly by studying the deviation of the coefficients of a from the coefficients of a0 and this is usually not a difficult task a more interesting application of this kind of approximation method is to elliptic homogenization problems for which we obtain results that give w 1 p estimates with a weak hypothesis of regularity in the coefficients for instance we are able to study the following cases

lipschitz estimates in almost periodic homogenization we establish uniform lipschitz estimates for second order elliptic systems in divergence form with rapidly oscillating almost periodic coefficients we give interior estimates as well as estimates up to the boundary in bounded c1 domains with either dirichlet or neumann data the main results extend those in the periodic setting due to avellaneda and lin for interior and dirichlet boundary estimates and later kenig lin and shen for the neumann boundary conditions in contrast to these papers our arguments are constructive and thus the constants are in principle computable and the results for the neumann conditions are new even in the periodic setting since we can treat nonsymmetric coefficients we also obtain uniform w1 p estimates 2015 wiley periodicals inc

an hp adaptive finite element method for electromagnetics part 3 a three dimensional infinite element for maxwell s equations this paper is a continuation of reference 26 cecot demkowicz and rachowicz computer methods in applied mechanics and engineering 2000 188 625 643 and describes an implementation of the infinite element for three dimensional time harmonic maxwell s equations proposed in reference 15 demkowicz and pal computer methods in applied mechanics and engineering 1998 164 77 94 the element is compatible with the hp finite element discretizations for maxwell s equations in bounded domains reported in references 16 18 computer methods in applied mechanics and engineering 1998 152 103 124 1999 169 331 344 2000 187 307 337 copyright 2003 john wiley sons ltd

convergence rates and w1 p estimates in homogenization theory of stokes systems in lipschitz domains abstract concerned with the stokes systems with rapidly oscillating periodic coefficients we mainly extend the recent works in 19 20 to those in term of lipschitz domains the arguments employed here are quite different from theirs and the basic idea comes from 37 originally motivated by 23 27 33 we obtain an almost sharp o e ln r 0 e convergence rate in l 2 space and a sharp o e error estimate in l 2 d d 1 space by a little stronger assumption under the dimensional condition d 2 we also establish the optimal o e convergence rate on pressure terms in l 2 space then utilizing the convergence rates we can derive the w 1 p estimates uniformly down to microscopic scale e without any smoothness assumption on the coefficients where 1 p 1 2 1 2 d and is a positive constant independent of e combining the local estimates based upon vmo coefficients consequently leads to the uniform w 1 p estimates here the proofs do not rely on the well known compactness methods

goal oriented adaptivity using unconventional error representations for the 1d helmholtz equation in this work the error of a given output functional is represented using bilinear forms that are different from those given by the adjoint problem these representations can be employed to design novel h p and h p energy norm and goal oriented adaptive algorithms numerical results in 1d show that for wave propagation problems the advantages of this new representation are notorious when selecting the laplace equation as the dual problem specifically the computed upper bounds of the new error representation are sharper than the classical ones used in both energy norm and goal oriented adaptive methods especially when the dispersion pollution error is significant

elliptic equations in divergence form with partially bmo coefficients the solvability in sobolev spaces is proved for divergence form second order elliptic equations in the whole space a half space and a bounded lipschitz domain for equations in the whole space or a half space the leading coefficients aij are assumed to be only measurable in one direction and have locally small bmo semi norms in the other directions for equations in a bounded domain additionally we assume that aij have small bmo semi norms in a neighborhood of the boundary of the domain we give a unified approach of both the dirichlet boundary problem and the conormal derivative problem we also investigate elliptic equations in sobolev spaces with mixed norms under the same assumptions on the coefficients

necessary and sufficient conditions for absolute continuity of elliptic harmonic measure our purpose in this paper is to give a necessary and sufficient condition on the modulus of continuity of the coefficients of an elliptic operator in divergence form in order that the corresponding poisson kernel for the dirichlet problem exist this condition on the coefficients is global continuity together with the additional property that the modulus of continuity along some nontangential direction at each boundary point be bounded uniformly in these points and directions by a function 7 t satisfying the dini type condition

goal oriented hp adaptivity for elliptic problems we propose and test a fully automatic goal oriented hp adaptive strategy for elliptic problems the method combines two techniques the standard goal oriented adaptivity based on a simultaneous solution of a dual problem and a recently proposed hp strategy based on minimizing the projection based interpolation error of a reference solution the proposed strategy is illustrated with two numerical examples laplace equation in l shape domain and an axisymmetric maxwell problem involving radiation of a loop antenna wrapped around a metallic cylinder into a conductive medium

elliptic equations with bmo coefficients in lipschitz domains in this paper we study inhomogeneous dirichlet problems for elliptic equations in divergence form optimal regularity requirements on the coefficients and domains for the w 1 p 1 p estimates are obtained the principal coefficients are supposed to be in the john nirenberg space with small bmo semi norms the domain is supposed to have lipschitz boundary with small lipschitz constant these conditions for the w 1 p theory do not just weaken the requirements on the coefficients they also lead to a more general geometric condition on the domain

homogenization of nonlinear equations with mixed boundary conditions in this paper we will study the convergence rates of solutions for homogenization of quasilinear elliptic equations with the mixed dirichlet robin boundary conditions thanks to the smoothing operator as well as homogenization tools we could handle the different boundary conditions in a uniform fashion as a consequence we establish the sharp rates of convergence in h1 and l2 which may be regarded as an extension from the classical linear equations dirichlet or neumann problems to a nonlinear case with the mixed boundary settings in this paper we will study the convergence rates of solutions for homogenization of quasilinear elliptic equations with the mixed dirichlet robin boundary conditions thanks to the smoothing operator as well as homogenization tools we could handle the different boundary conditions in a uniform fashion as a consequence we establish the sharp rates of convergence in h1 and l2 which may be regarded as an extension from the classical linear equations dirichlet or neumann problems to a nonlinear case with the mixed boundary settings

a self adaptive goal oriented hp finite element method with electromagnetic applications part ii electrodynamics abstract we present the formulation implementation and applications of a self adaptive goal oriented hp finite element fe method for electromagnetic em problems the algorithm delivers without any user interaction a sequence of optimal hp grids this sequence of grids minimizes the error in a prescribed quantity of interest with respect to the problem size and it converges exponentially in terms of the relative error in a user prescribed quantity of interest against the cpu time including problems involving high material contrasts boundary layers and or several singularities the goal oriented refinement strategy is an extension of a fully automatic energy norm based hp adaptive algorithm we illustrate the efficiency of the method with 2d numerical simulations of maxwell s equations using both h 1 conforming continuous elements and h curl conforming nedelec edge elements applications include alternate current ac resistivity logging instruments in a borehole environment with steel casing for the assessment of rock formation properties behind casing logging instruments steel casing and rock formation properties are assumed to exhibit axial symmetry around the axis of a vertical borehole for the presented challenging class of problems the self adaptive goal oriented hp fem delivers results with 5 7 digits of accuracy in the quantity of interest

a negative norm least squares method for time harmonic maxwell equations abstract this paper presents and analyzes a negative norm least squares finite element discretization method for the dimension reduced time harmonic maxwell equations in the case of axial symmetry the reduced equations are expressed in cylindrical coordinates and the analysis consequently involves weighted sobolev spaces based on the degenerate radial weighting the main theoretical results established in this work include existence and uniqueness of the continuous and discrete formulations and error estimates for simple finite element functions numerical experiments confirm the error estimates and efficiency of the method for piecewise constant coefficients

compactness methods in the theory of homogenization ii equations in non divergence form we prove c0 c1 and c1 1 a priori estimates for solutions of boundary value problems for elliptic operators with periodic coefficients of the form j 1ai j x 2 xi xj the constants in these estimates are independent of the small parameter and hence our results imply strengthened versions of the classical averaging theorems these results generalize to a wide class of linear operators in non divergence form including equations with lower order terms and nonuniformly oscillating coefficients as well as to certain nonlinear problems which we discuss in the last section

convergence rates and interior estimates in homogenization of higher order elliptic systems abstract this paper is concerned with the quantitative homogenization of 2m order elliptic systems with bounded measurable rapidly oscillating periodic coefficients we establish the sharp o e convergence rate in w m 1 p 0 with p 0 2 d d 1 in a bounded lipschitz domain in r d as well as the uniform large scale interior c m 1 1 estimate with additional smoothness assumptions the uniform interior c m 1 1 w m p and c m 1 estimates are also obtained as applications of the regularity estimates we establish asymptotic expansions for fundamental solutions

fourier series expansion in a non orthogonal system of coordinates for the simulation of 3d dc borehole resistivity measurements abstract we describe a new method to simulate 3d borehole resistivity measurements at zero frequency dc the method combines the use of a fourier series expansion in a non orthogonal system of coordinates with an existing 2d goal oriented higher order self adaptive hp finite element algorithm the new method is suitable for simulating measurements acquired with borehole logging instruments in deviated wells it delivers high accuracy simulations and it enables a considerable reduction of the computational complexity with respect to available 3d simulators since the number of fourier modes basis functions needed to solve practical applications is limited typically below 10 furthermore numerical results indicate that the optimal 2d grid based on the 0th fourier mode also called central fourier mode can be employed to efficiently solve the final 3d problem thereby avoiding the expensive construction of optimal 3d grids specifically for a challenging through casing resistivity application we reduce the computational time from several days using a 3d simulator to just 2 h with the new method while gaining accuracy the new simulation method can be easily extended to different physical phenomena with similar geometries as those arising in the simulation of 3d borehole electrodynamics and sonic acoustics coupled with elasticity measurements in addition the method is especially suited for inversion since we demonstrate that the number of fourier modes needed for the exact representation of the materials is limited to only one the central mode for the case of borehole measurements acquired in deviated wells

on goal oriented error estimation for elliptic problems application to the control of pointwise errors a method to estimate and control pointwise errors in finite element approximations of elliptic problems is presented as an application of the more general theory of goal oriented error estimation in the latter the accuracy of numerical approximations is assessed in terms of measures that are of practical interest to engineers rather than the classical global energy norm goal oriented error estimation requires the solution of a problem dual to the original problem and the computation of several global error estimates the performance of the method is carefully tested in the particular case of estimation and control of pointwise errors on a one dimensional problem

time domain goal oriented adaptivity using pseudo dual error representations abstract goal oriented adaptive algorithms produce optimal grids to solve challenging engineering problems recently a novel error representation using unconventional pseudo dual problems for goal oriented adaptivity in the context of frequency domain wave propagation problems has been developed in this paper we extend this error representation to the case of time domain problems we express the entire problem in weak form in order to derive the adjoint formulation and apply goal oriented adaptivity one dimensional 1d numerical results show that upper bounds for the new error representation are sharper than the classical ones therefore this new error representation can be used to design more efficient goal oriented adaptive methodologies

convergence for elliptic equations in periodic perforated domains abstract convergence for the solutions of elliptic equations in periodic perforated domains is concerned let denote the size ratio of the holes of a periodic perforated domain to the whole domain it is known that by energy method the gradient of the solutions of elliptic equations is bounded uniformly in in l 2 space also when approaches 0 the elliptic solutions converge to a solution of some simple homogenized elliptic equation in this work above results are extended to general w 1 p space for p 1 more precisely a uniform w 1 p estimate in for p 1 and a w 1 p convergence result for p n n 2 for the elliptic solutions in periodic perforated domains are derived here n is the dimension of the space domain one also notes that the l p norm of the second order derivatives of the elliptic solutions in general cannot be bounded uniformly in

integration of hp adaptivity and a two grid solver for electromagnetic problems we present implementation details and analyze convergence of a two grid solver forming the core of a fully automatic hp adaptive strategy for electromagnetic problems the solver delivers a solution for a fine grid obtained from an arbitrary coarse hp grid by a global hp refinement the classical v cycle algorithm combines an overlapping block jacobi smoother with optimal relaxation and a direct solve on the coarse grid a theoretical analysis of the two grid solver is illustrated with numerical experiments several electromagnetic applications show the efficiency of combining the fully automatic hp adaptive strategy with the two grid solver

a hierarchical duality approach to bounds for the outputs of partial differential equations abstract we present a technique for generating lower and upper bounds to outputs which are linear functionals of the solutions to finiteelement discretizations of symmetric or nonsymmetric coercive linear partial differential equations the method is based upon the construction of an augmented lagrangian which integrates i a quadratic energy reformulation of the desired output as the objective to be minimized with ii the finite element equilibrium equations and conforming hybridized intersubdomain continuity conditions as the constraints to be satisfied the bounds are then derived by appealing to the associated dual unconstrained max min problem evaluated for optimally chosen candidate lagrange multipliers generated by a less expensive approximation such as a low dimensional finite element discretization as in many a posteriori error estimation techniques the bound calculation requires only the solution of subdomain local symmetric neumann problems on the refined truth mesh the technique is presented and illustrated for the case of the one dimensional convection diffusion equation

the l p regularity problem for the stokes system on lipschitz domains we study the lp regularity problem for the stationary stokes system on lipschitz domains for any p 2 we show that a weak reverse holder inequality with exponent p is both necessary and sufficient for the solvability of the regularity problem with data in w1 p lnp we also obtain the w1 p estimate u lp q lp f lp in a bounded lipschitz domain rd for solutions to the dirichlet problem u q div f div u 0 in and u 0 on where 1p 12 12d e