regularity of solutions of elliptic problems with a curved fracture abstract we consider the laplace equation with a right hand side concentrated on a curved fracture of class c m 2 for some nonnegative integer m i e a sort of dirac mass we show that the solution belongs to a weighted sobolev space of order m the weight being the distance to this fracture our proof relies on a priori estimates in a dihedron or a cone with singularities for elliptic operators with variable coefficients in both cases such an estimate is obtained using a dyadic covering of the domain

a new paradigm for parallel adaptive meshing algorithms we present a new approach to the use of parallel computers with adaptive finite element methods this approach addresses the load balancing problem in a new way requiring far less communication than current approaches it also allows existing sequential adaptive pde codes such as pltmg and mc to run in a parallel environment without a large investment in recoding in this new approach the load balancing problem is reduced to the numerical solution of a small elliptic problem on a single processor using a sequential adaptive solver without requiring any modifications to the sequential solver the small elliptic problem is used to produce a posteriori error estimates to predict future element densities in the mesh which are then used in a weighted recursive spectral bisection of the initial mesh the bulk of the calculation then takes place independently on each processor with no communication using possibly the same sequential adaptive solver each processor adapts its region of the mesh independently and a nearly load balanced mesh distribution is usually obtained as a result of the initial weighted spectral bisection only the initial fan out of the mesh decomposition to the processors requires communication two additional steps requiring boundary exchange communication may be employed after the individual processors reach an adapted solution namely the construction of a global conforming mesh from the independent subproblems followed by a final smoothing phase using the subdomain solutions as an initial guess we present a series of convincing numerical experiments which illustrate the effectiveness of this approach the justification of the initial refinement prediction step as well as the justification of skipping the two communication intensive steps is supported by some recent j xu and a zhou math comp to appear and not so recent j a nitsche and a h schatz math comp 28 1974 pp 937 958 a h schatz and l b wahlbin math comp 31 1977 pp 414 442 a h schatz and l b wahlbin math comp 64 1995 pp 907 928 results on local a priori and a posteriori error estimation

finite element discretization of state constrained elliptic optimal control problems with semilinear state equation we study a class of semilinear elliptic optimal control problems with pointwise state constraints the purpose of this paper is twofold first we present convergence results for the finite element discretization of this problem class similarly to known results with finite dimensional control space thus extending results that are for control functions only available for linear quadratic convex problems we rely on a quadratic growth condition for the continuous problem that follows from second order sufficient conditions second we show that the second order sufficient conditions for the continuous problem transfer to its discretized version this is of interest for example when considering questions of local uniqueness of solutions or the convergence of solution algorithms such as the sqp method

a priori error estimates for three dimensional parabolic optimal control problems with pointwise control in this paper we provide an a priori error analysis for parabolic optimal control problems with a pointwise dirac type control in space on three dimensional domains the two dimensional case was treated in d leykekhman and b vexler siam j numer anal 51 2013 pp 2797 2821 however the three dimensional case is technically much more involved to approximate the problem we use standard continuous piecewise linear elements in space and the piecewise constant discontinuous galerkin method in time despite low regularity of the state equation we establish mathcal o sqrt k h order of convergence rate for the control in the l 2 norm this result improves almost twice the previously known estimate in w gong m hinze and z zhou siam j control optim 52 2014 pp 97 119 and in addition does not require any relationship between the time step k and the mesh size h the main technical tools are discrete maximal parabolic regularity results and the best approximation type estim

piecewise polynomial interpolation in muckenhoupt weighted sobolev spaces and applications we develop a constructive piecewise polynomial approximation theory in weighted sobolev spaces with muckenhoupt weights for any polynomial degree the main ingredients to derive optimal error estimates for an averaged taylor polynomial are a suitable weighted poincare inequality a cancellation property and a simple induction argument we also construct a quasi interpolation operator built on local averages over stars which is well defined for functions in l 1 l1 we derive optimal error estimates for any polynomial degree on simplicial shape regular meshes on rectangular meshes these estimates are valid under the condition that neighboring elements have comparable size which yields optimal anisotropic error estimates over n n rectangular domains the interpolation theory extends to cases when the error and function regularity require different weights we conclude with three applications nonuniform elliptic boundary value problems elliptic problems with singular sources and fractional powers of elliptic operators

quadratic forms weighted eigenfunctions and boundary value problems for non linear second order ordinary differential equations some known results for different kinds of boundary value problems for second order ordinary differential equations are generalised different approaches are compared with one another using topological and variational methods and the theory of weighted eigenvalue problems

l2 estimates for the finite element method for the dirichlet problem with singular data in this paper we consider the approximation by the finite element method of second order elliptic problems on convex domains and homogeneous dirichlet condition on the boundary in these problems the data are borel measures using a quasiuniform mesh of finite elements and polynomials of degree 1 we prove that in two dimensions the convergence is of orderh inl 2 and in three dimensions of orderh 1 2

a posteriori error estimates for elliptic problems with dirac delta source terms the aim of this paper is to introduce residual type a posteriori error estimators for a poisson problem with a dirac delta source term in l p norm and w1 p seminorm the estimators are proved to yield global upper and local lower bounds for the corresponding norms of the error they are used to guide adaptive procedures which are experimentally shown to lead to optimal orders of convergence

c0 interior penalty methods for an elliptic state constrained optimal control problem with neumann boundary condition abstract we study c 0 interior penalty methods for an elliptic optimal control problem with pointwise state constraints on two dimensional convex polygonal domains the approximation of the optimal state is obtained by solving a fourth order variational inequality and the approximation of the optimal control is computed by a post processing procedure we prove the convergence of numerical solutions with rates in the h 2 like energy error by using the complementarity form of the variational inequality furthermore we develop an a posteriori analysis for a residual based error estimator and introduce an adaptive algorithm numerical experiments are provided to gauge the performance of the proposed methods

force traction microscopy an inverse problem with pointwise observations abstract force traction microscopy is an inversion method that allows one to obtain the stress field applied by a living cell on the environment on the basis of a pointwise knowledge of the displacement produced by the cell itself this classical biophysical problem usually addressed in terms of green functions can be alternatively tackled using a variational framework and then a finite elements discretization in such a case a variation of the error functional under suitable regularization is operated in view of its minimization this setting naturally suggests the introduction of a new equation based on the adjoint operator of the elasticity problem in this paper we illustrate the rigorous theory of the two dimensional and three dimensional problems involving in the former case a distributed control and in the latter case a surface control the pointwise observations require one to exploit the theory of elasticity extended to forcing terms that are borel measures

sharply local pointwise a posteriori error estimates for parabolic problems we prove pointwise a posteriori error estimates for semi and fully discrete finite element methods for approximating the solution u to a parabolic model problem our estimates may be used to bound the finite element error u u h l d where d is an arbitrary subset of the space time domain of the definition of the given pde in contrast to standard global error estimates these estimators de emphasize spatial error contributions from space time regions removed from d our results are valid on arbitrary shape regular simplicial meshes which may change in time and also provide insight into the contribution of mesh change to local errors when implemented in an adaptive method these estimates require only enough spatial mesh refinement away from d in order to ensure that local solution quality is not polluted by global effects

a priori error analysis for discretization of sparse elliptic optimal control problems in measure space in this paper an optimal control problem is considered where the control variable lies in a measure space and the state variable fulfills an elliptic equation this formulation leads to a sparse structure of the optimal control in this setting we prove a new regularity result for the optimal state and the optimal control moreover a finite element discretization based on e casas c clason and k kunisch siam j control optim 50 2012 pp 1735 1752 is discussed and a priori error estimates are derived which significantly improve the estimates from that paper numerical examples for problems in two and three space dimensions illustrate our results

eigenfunction methods and nonlinear hyperbolic boundary value problems at resonance the abstract boundary value problem lu gu f u dom l h is considered here h is used to denote a real separable hilbert space l a closed symmetric linear operator and g a nonlinear operator assumed to be lipschitz continuous and strongly monotone in addition l is assumed to have a complete set of eigenfunctions in h and is allowed to have an infinite dimensional null space the existence of unique solutions depending continuously on f is established by a constructive approach galerkin approximations are considered and error estimates are given as an application of the main result the existence of time periodic weak solutions of the n dimensional wave equation is shown

some optimal error estimates for piecewise linear finite element approximations it is shown that the ritz projection onto spaces of piecewise linear finite elements is bounded in the sobolev space wl for 2 p oc this implies that for functions in wf n w2 the error in approximation behaves like 0 h in wp for 2 p oo and like 0 h2 in lp for 2 p oo in all these cases the additional logarithmic factor previously included in error estimates for linear finite elements does not occur

adaptive finite element method for parabolic equations with dirac measure abstract in this paper we study the adaptive finite element method for parabolic equations with dirac measure two kinds of problems with separate measure data in time and measure data in space are considered it is well known that the solutions of such kind of problems may exhibit lower regularity due to the existence of the dirac measure and thus fit to adaptive fem for space discretization and variable time steps for time discretization for both cases we use piecewise linear and continuous finite elements for the space discretization and backward euler scheme or equivalently piecewise constant discontinuous galerkin method for the time discretization the a posteriori error estimates based on energy and l 2 norms for the fully discrete problems are then derived to guide the adaptive procedure numerical results are provided at the end of the paper to support our theoretical findings

optimal a priori error estimates for an elliptic problem with dirac right hand side it is well known that finite element solutions for elliptic pdes with dirac measures as source terms converge due to the fact that the solution is not in h 1 suboptimal in classical norms a standard remedy is to use graded meshes then quasioptimality i e optimal up to a log factor for low order finite elements can be recovered e g in the l 2 norm here we show for the lowest order case quasioptimality and for higher order finite elements optimal order a priori estimates on a family of quasi uniform meshes in an l 2 seminorm the seminorm is defined as an l 2 norm on a fixed subdomain which excludes the locations of the delta source terms our motivation in the use of such a norm results from the observation that in many applications the error at the singularity is dominated by the model error e g in dimension reduced settings or is not the quantity of interest e g in optimal control problems the quasi optimal and optimal order a priori bounds are obtained recursively by using aub

a priori error analysis for finite element approximation of parabolic optimal control problems with pointwise control we consider finite element approximations of parabolic control problems with pointwise control the state equation exhibits low regularity due to the control imposed pointwisely this introduces some difficulties for both theoretical and numerical analysis to discretize the optimal control problem we use variational discretization together with piecewise linear and continuous finite elements for the space discretization of the state and we use the backward euler scheme for time discretization we prove a priori error estimates for the control state and adjoint state numerical experiments are provided which support the theoretical results

interior maximum norm estimates for finite element methods part ii we consider bilinear forms a connected with second order elliptic problems and assume that for u h in a finite element space s h we have a u u h x f x for x in s h with local compact support we give local estimates for u u h in l and w 1 of the type local best approximation plus weak outside influences plus the local size of f

computing class fields via the artin map based on an explicit representation of the artin map for kummer extensions we present a method to compute arbitrary class fields as in the proofs of the existence theorem the problem is first reduced to the case where the field contains sufficiently many roots of unity using kummer theory and an explicit version of the artin reciprocity law we show how to compute class fields in this case we conclude with several examples

computing generators of free modules over orders in group algebras abstract let e be a number field and g be a finite group let a be any o e order of full rank in the group algebra e g and x be a left a lattice we give a necessary and sufficient condition for x to be free of given rank d over a in the case that the wedderburn decomposition e g m is explicitly computable and each m is in fact a matrix ring over a field this leads to an algorithm that either gives elements 1 d x such that x a 1 a d or determines that no such elements exist let l k be a finite galois extension of number fields with galois group g such that e is a subfield of k and put d k e the algorithm can be applied to certain galois modules that arise naturally in this situation for example one can take x to be o l the ring of algebraic integers of l and a to be the associated order a e g o l e g the application of the algorithm to this special situation is implemented in magma under certain extra hypotheses when k e q

a priori mesh grading for an elliptic problem with dirac right hand side the green function of the poisson equation in two dimensions is not contained in the sobolev space h 1 omega such that finite element error estimates for the discretization of a problem with the dirac measure on the right hand side are nonstandard and quasi uniform meshes are inappropriate by using graded meshes l 2 error estimates of almost optimal order are shown as a byproduct we show for the poisson equation with a right hand side in l 2 that appropriate mesh refinement near some interior point diminishes the error at this point by nearly one order

control of an elliptic problem with pointwise state constraints this paper deals with a quadratic control problem for elliptic equations with pointwise state constraints existence and uniqueness of the solution is proved optimality conditions are given and regularity of the optimal solution is investigated

a posteriori error estimates for the problem of electrostatics with a dipole source abstract electroencephalography is a non invasive technique for detecting brain activity from the measurement of the electric potential on the head surface in mathematical terms it reduces to an inverse problem in which the goal is to determine the source that has generated the electric field from measurements of boundary values of the electric potential since for reasonable models the time variation of the electric and magnetic fields can be disregarded the mathematical modeling of the corresponding forward problem leads to an electrostatics problem with a current dipole source this is a singular problem since the current dipole model involves first order derivatives of a dirac delta measure its solution lies in l p for 1 p 3 2 in three dimensional domains and 1 p 2 in the two dimensional case we consider the numerical approximation of the forward problem by means of standard piecewise linear continuous finite elements we prove a priori error estimates in the l p norm then we propose a residual type a posteriori error estimator we prove that it is reliable and efficient namely it yields global upper and local lower bounds for the corresponding norms of the error finally we use this estimator to guide an adaptive procedure which is experimentally shown to lead to an optimal order of convergence

writing representations over proper division subrings abstract let e be a division ring and g a finite group of automorphisms of e whose elements are distinct modulo inner automorphisms of e let f e g be the division subring of elements of e fixed by g given a representation a e d d of an f algebra a we give necessary and sufficient conditions for to be writable over f here e d d denotes the algebra of d d matrices over e and a matrix a writes over f if a 1 a a f d d we give an algorithm for constructing an a or proving that no a exists the case of particular interest to us is when e is a field and is absolutely irreducible the algorithm relies on an explicit formula for a and a generalization of hilbert s theorem 90 that arises in galois cohomology the algorithm has applications to the construction of absolutely irreducible group representations especially for solvable groups and to the recognition of class c 5 in aschbacher s matrix group classification scheme m aschbacher on the maximal subgroups of the finite classical groups invent math 76 1984 469 514 mr0746539 shangzhi li on the subgroup structure of classical groups in group theory in china in math appl china ser vol 365 kluwer acad publ dordrecht 1996 pp 70 90 mr1447199 1 13

local error analysis for the stokes equations with a punctual source term the solution of the stokes problem with a punctual force in source term is not h 1 times mathbb l 2 and therefore the approximation by a finite element method is suboptimal in the case of poisson problem with a dirac mass in the right hand side an optimal convergence for the lagrange finite elements has been shown on a subdomain which excludes the singularity in mathbb l 2 norm by koppl and wohlmuth here we show a quasi optimal local convergence in h 1 times mathbb l 2 norm for a mathbb p k mathbb p k 1 finite element method k geqslant 2 and for the mathbb p 1 mathrm b mathbb p 1 the error is still analysed on a subdomain which does not contain the singularity the proof is based on local arnold and liu error estimates a weak version of aubin nitsche duality lemma applied to the stokes problem and discrete inf sup conditions these theoretical results are generalized to a wide class of finite element methods before being illustrated by numerical simulations

local and parallel finite element algorithms based on two grid discretization for the stream function form of navier stokes equations abstract a new local and parallel discretization finite element algorithms are proposed and analyzed in this paper for the stream function form of navier stokes equations according to 1 the theoretical tools for analyzing these methods are some local priori estimate that are also obtained in this paper for finite element solutions on general shape regular grids some numerical experiments are also presented to support the theory

an adaptive stabilized finite element scheme for a water quality model abstract residual type a posteriori error estimators are introduced in this paper for an advection diffusion reaction problem with a dirac delta source term the error is measured in an adequately weighted w 1 p norm these estimators are proved to yield global upper and local lower bounds for the corresponding norms of the error they are used to guide adaptive procedures which are experimentally shown to lead to optimal orders of convergence

eine aussage zurl stabilitat und zur genauen konvergenzordnung derh 0 0 projektionen in this paper we settle a question concerning the necessity of the logh factor appearing in error estimates of linear finite element solutions let 2 be a bounded convex polygon and t h a regular family of subdivisions of into triangles be furthers 0 h h 0 1 the corresponding linear finite element spaces andp h h 0 1 s 0 h theh 0 1 projections with respect to the uniform norm asymptotically estimates of the form parallel u p h u parallel infty leqq c logh dist u s 0 h u in c bar omega cap h 0 1 omega hold as pointed out by schatz 1980 for functionsu with bounded second derivatives this implies parallel u p h u parallel infty leqq ch 2 log h u 2 infty a result which was proved by nitsche 1975 and scott 1976 with different methods it was an open question if the factor logh is necessary or only due to the methods of proof in this paper it is shown by an example that the cited estimates are sharp in the very strong sense that in general they are no longer valid if logh is replaced by a termo logh

robust error estimates for the finite element approximation of elliptic optimal control problems in this paper we consider the finite element approximation of an elliptic optimal control problem based on an assumption on the adjoint state of the continuous problem with a small parameter which represents a regularization of the bang bang type control problem we derive robust a priori error estimates for optimal control and state and a posteriori error estimate is also presented numerical experiments confirm our theoretical results

existence of periodic solutions for nonlinear differential equations abstract in this paper we shall study the existence of a periodic solution to the nonlinear differential equationxtt axt a xt a ax b t x f t x t 0 in some complex hilbert space using duality and variational methods

approximation of elliptic control problems in measure spaces with sparse solutions optimal control problems in measure spaces governed by elliptic equations are considered for distributed and neumann boundary control which are known to promote sparse solutions optimality conditions are derived and some of the structural properties of their solutions in particular sparsity are discussed a framework for their approximation is proposed which is efficient for numerical computations and for which we prove convergence and provide error estimates

adaptive finite element methods for an optimal control problem involving dirac measures the purpose of this work is the design and analysis of a reliable and efficient a posteriori error estimator for the so called pointwise tracking optimal control problem this linear quadratic optimal control problem entails the minimization of a cost functional that involves point evaluations of the state thus leading to an adjoint problem with dirac measures on the right hand side control constraints are also considered the proposed error estimator relies on a posteriori error estimates in the maximum norm for the state and in muckenhoupt weighted sobolev spaces for the adjoint state we present an analysis that is valid for two and three dimensional domains we conclude by presenting several numerical experiments which reveal the competitive performance of adaptive methods based on the devised error estimator

finite element methods in local active control of sound the active control of sound is analyzed in the framework of the mathematical theory of optimal control after setting the problem in the frequency domain we deal with the state equation which is a helmholtz partial differential equation we show the existence of a unique solution and analyze a finite element approximation when the source term is a dirac delta measure two optimization problems are successively considered the first one concerns the choice of phases and amplitudes of the actuators to minimize the noise at the sensors location the second one consists of determining the optimal actuators placement both problems are then numerically solved error estimates are settled and numerical results for some tests are reported

approximations of elliptic optimal control problems with controls acting on a lower dimensional manifold in this paper we study finite element approximations to some elliptic optimal control problems with controls acting on a lower dimensional manifold which can be a point a curve or a surface we use piecewise linear finite elements to approximate state variables while utilizing the variational discretization to approximate control variables we derive several a priori error estimates for optimal controls from different cases depending on the dimensions of the computational domain and the manifold where controls act we ends up with extensive numerical experiments which confirm our theoretical findings

an a posteriori error analysis of an elliptic optimal control problem in measure space abstract we propose an a posteriori error estimator for an optimal control problem with sparsity promoting cost the control variable lies in the space of regular borel measures we consider a solution technique that relies on the discretization of the control variable as a linear combination of dirac measures the proposed a posteriori error estimator can be decomposed into the sum of two contributions an error estimator in the maximum norm for the discretization of the adjoint equation and an estimator in the l 2 norm that accounts for the approximation of the state equation we prove that the designed error estimator is locally efficient and we explore its reliability properties the analysis is valid for two and three dimensional domains we illustrate the theory with numerical examples

on periodic solutions of nonlinear second order vector differential equations this paper considers existence of periodic solutions for vector lienard differential equations in our main result we write where q t x is a symmetric matrix and h t x is sublinear the key assumption relates the asymptotic behaviour as x of the eigenvalues of q t x to the spectrum of the linear operator d 2 dt 2 several choices for q t x are considered which lead to known theorems and extend others in the case of the duffing equation the assumptions are weakened our approach is based on leray schauder s degree theory and a priori estimates

a posteriori error analysis for a class of integral equations and variational inequalities we consider elliptic and parabolic variational equations and inequalities governed by integro differential operators of order 2s in 0 2 our main motivation is the pricing of european or american options under levy processes in particular pure jump processes or jump diffusion processes with tempered stable processes the problem is discretized using piecewise linear finite elements in space and the implicit euler method in time we construct a residual type a posteriori error estimator which gives a computable upper bound for the actual error in h s norm the estimator is localized in the sense that the residuals are restricted to the discrete non contact region numerical experiments illustrate the accuracy of the space and time estimators and show that they can be used to measure local errors and drive adaptive algorithms

maximum norm stability and maximal l p lp regularity of fems for parabolic equations with lipschitz continuous coefficients in this paper we study the semi discrete galerkin finite element method for parabolic equations with lipschitz continuous coefficients we prove the maximum norm stability of the semigroup generated by the corresponding elliptic finite element operator and prove the space time stability of the parabolic projection onto the finite element space in l infty q t l qt and l p 0 t l q omega lp 0 t lq 1

the optimal convergence of finite element methods for the three dimensional second order elliptic boundary value problem with singularity abstract let x 0 assume that g x 0 x satisfies the elliptic boundary value problem l g x 0 x x j a i j x g x 0 x x i x x 0 in g x 0 x 0 on where 3 is a convex polyhedral domain in this article we propose specially graded meshes t h furthermore we obtain an l 2 error estimate of order h k 1 ln h 3 2 for approximations with a p k finite element method

optimal a priori error estimates of parabolic optimal control problems with pointwise control in this paper we consider a parabolic optimal control problem with a pointwise dirac type control in space but variable in time in two space dimensions to approximate the problem we use the standard continuous piecewise linear approximation in space and the piecewise constant discontinuous galerkin method in time despite low regularity of the state equation we show almost optimal h 2 k convergence rate for the control in l 2 norm this result improves almost twice the previously known estimate in w gong m hinze and z zhou a priori error analysis for finite element approximation of parabolic optimal control problems with pointwise control tech report 2011 07 hamburger beitrage zur angewandten mathematik hamburg germany 2011

finite element approximation of elliptic problems with dirac measure terms in weighted spaces applications to one and three dimensional coupled problems in this work we study the stability and the convergence rates of the finite element approximation of elliptic problems involving dirac measures using weighted sobolev spaces and weighted discrete norms our approach handles both the cases where the measure is simply a right hand side and where it represents an additional term i e solution dependent in the formulation of the problem the main motivation of this study is to provide a methodological tool to treat elliptic problems in fractured domains where the coupling terms are seen as dirac measures concentrated on the fractures we first establish a decomposition lemma which is our fundamental tool for the analysis of the considered problems in the nonstandard setting of weighted spaces then we consider the stability of the galerkin approximation with finite elements in weighted norms with uniform and graded meshes we introduce a discrete decomposition lemma that extends the continuous one and allows us to derive discrete inf sup conditions in weighted norms then we focus on the convergence of the finite element method due to the lack of regularity the convergence rates are suboptimal for uniform meshes we show that using graded meshes optimal rates are recovered our theoretical results are supported by several numerical experiments finally we show how our theoretical results apply to certain coupled problems involving fluid flow in porous three dimensional media with one dimensional fractures that are found in the analysis of microvascular flows

a posteriori error estimates for optimal control problems with state and control constraints we discuss the full discretization of an elliptic optimal control problem with pointwise control and state constraints we provide the first reliable a posteriori error estimator that contains only computable quantities for this class of problems moreover we show that the error estimator converges to zero if one has convergence of the discrete solutions to the solution of the original problem the theory is illustrated by numerical tests

implicit explicit multistep methods for quasilinear parabolic equations efficient combinations of implicit and explicit multistep methods for nonlinear parabolic equations were recently studied in 1 in this note we present a refined analysis to allow more general nonlinearities the abstract theory is applied to a quasilinear parabolic equation

finite element approximation of dirichlet boundary control for elliptic pdes on two and three dimensional curved domains we consider the variational discretization of elliptic dirichlet optimal control problems with constraints on the control the underlying state equation which is considered on smooth two and three dimensional domains is discretized by linear finite elements taking into account domain approximation the control variable is not discretized we obtain optimal error bounds for the optimal control in two and three space dimensions and prove a superconvergence result in two dimensions provided that the underlying mesh satisfies some additional condition we confirm our analytical findings by numerical experiments

minimizing representations over number fields ii computations in the brauer group abstract finding minimal fields of definition for representations of finite groups is one of the most important unsolved problems of computational representation theory while good methods exist for representations over finite fields it is still an open question in the case of number fields continuing and extending previous work we give a practical method for finding defining fields of minimal degree for absolutely irreducible representations the method is based on techniques from galois cohomology and the use of an explicit form of a weak grunwald wang theorem

superconvergence in finite element methods and meshes that are locally symmetric with respect to a point consider a second order elliptic boundary value problem in any number of space dimensions with locally smooth coefficients and solution consider also its numerical approximation by standard conforming finite element methods with for example fixed degree piecewise polynomials on a quasi uniform mesh family the h method it will be shown that if the finite element function spaces are locally symmetric about a point x 0 with respect to the antipodal map x to x 0 x x 0 then superconvergence ensues at xo under mild conditions on what happens outside a neighborhood of x 0 for piecewise polynomials of even degree superconvergence occurs in function values for piecewise polynomials of odd degree it occurs in derivatives