the optimal rate of convergence of the p version of the boundary element method in two dimensions we introduce the jacobi weighted besov and sobolev spaces in the one dimensional setting in the framework of these spaces we analyze lower and upper bounds for approximation errors in the p version of the boundary element method for hypersingular and weakly singular integral operators on polygons we prove the optimal rate of convergence for the p version in the energy norms of and respectively

de rham diagram for hp finite element spaces abstract we prove that the hp finite elements for h curl spaces introduced in 1 fit into a general de rham diagram involving hp approximations the corresponding interpolation operators generalize the notion of hp interpolation introduced in 2 and are different from the classical operators of nedelec and raviart thomas

the optimal convergence rate of the p version of the finite element method optimal error estimates for the p version of the finite element method in two dimensions are proven for the case when u in h k omega or u has singularities induced by the corners of the domain the case of nonhomogenous essential boundary conditions is also analyzed

direct and inverse approximation theorems for the p version of the finite element method in the framework of weighted besov spaces part i approximability of functions in the weighted besov spaces this is the first of a series devoted to the approximation theory of the p version of the finite element method in two dimensions in the framework of the jacobi weighted besov spaces which provides the p version with a solid mathematical foundation in this paper we establish a mathematical framework of the jacobi weighted besov and sobolev spaces and analyze the approximability of the functions in the framework of these spaces particularly singular functions of r gamma type and r gamma log nu r type these spaces and the corresponding approximation properties are of fundamental importance to the proof of the optimal convergence for the p version in two dimensions in part ii and to various sharp inverse approximation theorems in part iii

gegenbauer approximation and its applications to differential equations on the whole line abstract a gegenbauer approximation is discussed several imbedding inequalities and inverse inequalities are obtained some approximation results are given by variable transformation differential equations on the whole line are changed to certain equations on a finite interval gegenbauer polynomials are used for their numerical solutions the stabilities and convergences of proposed schemes are proved the main idea and techniques used in this paper are also applicable to other multiple dimensional problems in unbounded domains

hp version discontinuous galerkin methods for hyperbolic conservation laws abstract the development of hp version discontinuous galerkin methods for hyperbolic conservation laws is presented in this work a priori error estimates are derived for a model class of linear hyperbolic conservation laws these estimates are obtained using a new mesh dependent norm that reflects the dependence of the approximate solution on the local element size and the local order of approximation the results generalize and extend previous results on mesh dependent norms to hp version discontinuous galerkin methods a posteriori error estimates which provide bounds on the actual error are also developed in this work numerical experiments verify the a priori estimates and demonstrate the effectiveness of the a posteriori estimates in providing reliable estimates of the actual error in the numerical solution

hp adaptive finite elements in electromagnetics a model problem for the steady state form of maxwell s equations is considered the problem is formulated in a weak form using a lagrange multiplier laying down a foundation for a general class of novel hp adaptive fe approximations a convergence proof for curvilinear elements is presented the proposed method is illustrated and verified by a series of 2d experiments which include elements with curved boundaries and nonhomogeneous media

on the homological dimensions of leavitt path algebras with coefficients in commutative rings abstract in this paper we give sharp bounds for the homological dimensions of the leavitt path algebra l k e of a finite graph e with coefficients in a commutative ring k as well as establish a formula for calculating the homological dimensions of l k e when k is a commutative unital algebra over a field

the hp mitc finite element method for the reissner mindlin plate problem the popular mitc finite elements used for the approximation of the reissner mindlin plate are extended to the case where elements of non uniform degree p distribution are used on locally refined meshes such an extension is of particular interest to the hp version and hp adaptive finite element methods a priori error bounds are provided showing that the method is locking free the analysis is based on new approximation theoretic results for non uniform brezzi douglas fortin marini spaces and extends the results obtained in the case of uniform order approximation on globally quasi uniform meshes presented by stenberg and suri siam j numer anal 34 1997 544 numerical examples illustrating the theoretical results and comparing the performance with alternative standard galerkin approaches are presented for two new benchmark problems with known analytic solution including the case where the shear stress exhibits a boundary layer the new method is observed to be locking free and able to provide exponential rates of convergence even in the presence of boundary layers

jacobi approximations in certain hilbert spaces and their applications to singular differential equations jacobi approximations in certain hilbert spaces are investigated several weighted inverse inequalities and poincare inequalities are obtained some approximation results are given singular differential equations are approximated by using jacobi polynomials this method keeps the spectral accuracy some linear problems and a nonlinear logistic equation are considered the stabilities and the convergences of proposed schemes are proved strictly the main idea and techniques used in this paper are also applicable to other singular problems in multiple dimensional spaces

the optimal convergence of the h p version of the finite element method with quasi uniform meshes in the framework of the jacobi weighted besov spaces we analyze the convergence of the h p version of finite element solutions on quasi uniform meshes and the lower and upper bounds of errors for elliptic problems on polygons both lower and upper bounds are proved to be optimal in h and p which leads to the optimal convergence of the h p version of the finite element method with quasi uniform meshes for elliptic problems on polygons the results proved for the h p version include the h version with quasi uniform meshes and the p version with quasi uniform degrees as two special cases

hp vector finite element method for eigenmode analysis of waveguides a new stable variational formulation for full wave analysis of waveguides is presented numerical experiments show that the method remains stable at very low frequencies theoretical results apply to a wide class of vector finite elements including two families of nedelec numer math 35 1980 315 numer math 50 1986 57 and their generalization hp edge elements comput methods appl mech engrg 152 1998 103 comput methods appl mech engrg 169 1999 331

uniform hp convergence results for the mortar finite element method the mortar finite element is an example of a non conforming method which can be used to decompose and re compose a domain into sub domains without requiring compatibility between the meshes on the separate components we obtain stability and convergence results for this method that are uniform in terms of both the degree and the mesh used without assuming quasiuniformity for the meshes our results establish that the method is optimal when non quasiuniform h or hp methods are used such methods are essential in practice for good rates of convergence when the interface passes through a corner of the domain we also give an error estimate for when the p version is used numerical results for hp and hp mortar fems show that these methods behave as well as conforming fems an hp extension theorem is also proved

hp discontinuous galerkin time stepping for parabolic problems abstract we consider the hp version of the discontinuous galerkin dg time stepping method for linear parabolic problems with non symmetric elliptic spatial operators we derive new analyticity estimates for the exact solutions by means of semigroup techniques these estimates allow us to show that the hp dg time stepping method can resolve start up singularities at exponential rates of convergence

optimal estimates for lower and upper bounds of approximation errors in the p version of the finite element method in two dimensions this paper will analyze the lower and upper error bounds of the finite element solution of the p version for linear elliptic problems in polygonal domains the optimal rate of convergence is rigorously proved based on the sharp estimates of lower and upper bounds of the approximation error

convergence analysis of an energy based discontinuous galerkin method for the wave equation in second order form hp version abstract in this paper we study the error estimates of the h p version of a spatial discontinuous galerkin method proposed in appelo and hagstrom 2015 for the wave equation for different degrees of the approximation spaces we give the error estimates depending on the flux parameters in the energy norm and prove that optimal convergence rates can be obtained for some special flux parameters which is an improvement of the existing work of appelo and hagstrom 2015 finally we provide numerical experiments to verify our theoretical findings

mixed hp finite element methods for stokes and non newtonian flow abstract we analyze the stability of hp finite elements for viscous incompressible flow for the classical velocity pressure formulation we give new estimates for the discrete inf sup constants on geometric meshes which are explicit in the polynomial degree k of the elements in particular we obtain new bounds for p elements on triangles for the three field stokes problem describing linearized non newtonian flow we estimate discrete inf sup constants explicit in both h and k for various subspace choices continuous and discontinuous for the extra stress we also give a stability analysis of the hp version of an elastic viscous split stress evss method and present elements that are stable and optimal in h and k finally we present numerical results that show the exponential convergence of the hp version for stokes flow over unsmooth domains

mixed hp dgfem for incompressible flows we consider several mixed discontinuous galerkin approximations of the stokes problem and propose an abstract framework for their analysis using this framework we derive a priori error estimates for hp approximations on tensor product meshes we also prove a new stability estimate for the discrete divergence bilinear form