on the finite volume element method for general self adjoint elliptic problems the finite volume element method fve is a discretization technique for partial differential equations this paper develops discretization energy error estimates for general self adjoint elliptic boundary value problems with fve based on triangulations on which there exist linear finite element spaces and a very general type of control volumes covolumes the energy error estimates of this paper are also optimal but the restriction conditions for the covolumes given in r e bank and d j rose siam j numer anal 24 1987 pp 777 787 z q cai numer math 58 1991 pp 713 735 are removed the authors finally provide a counterexample to show that an expected l2 error estimate does not exist in the usual sense it is conjectured that the optimal order of u u h 0 omega should be o h for the general case

mixed boundary value problems for elliptic and parabolic differential equations of second order on considere les problemes aux valeurs limites mixtes suivants lu f 1 dans mu f 2 sur u f 3 sur avec l operateur elliptique et pu f 1 dans 0 t mu f 2 sur 0 t u f 3 sur 0 t 0 pour des operateurs p paraboliques pour des operateurs obliques generaux m et pour des domaines ou et se rencontrent a des angles arbitraires on montre l existence de solutions classiques quand les angles sont non nuls et f 3 est continue de holder

duality for ext groups and extensions of discrete series for graded hecke algebras in this paper we study extensions of graded affine hecke algebra modules in particular based on an explicit projective resolution on graded affine hecke algebra modules we prove a duality result for ext groups this duality result with an ind res resolution gives an algebraic proof of the fact that all higher ext groups between a tempered module and a discrete series vanish

on extensions of generalized steinberg representations abstract let f be a local non archimedean field and let g be the group of f valued points of a connected reductive algebraic group over f in this paper we compute the ext groups of generalized steinberg representations in the category of smooth g representations with coefficients in a certain self injective ring

the finite volume element method for diffusion equations on general triangulations this paper develops discretization error estimates for the finite volume element method on general triangulations of a polygonal domain in mathcal r 2 using a special type of control volume the theory applies to diffusion equations of the form begin gathered nabla a nabla u f quad text in omega hfill u 0 quad text on partial omega hfill end gathered under fairly general conditions the theory establishes o h estimates of the error in a discrete mathcal h 1 seminorm under an additional assumption concerning local uniformity of the triangulation the estimate is improved to o h 2

the cohomology of period domains for reductive groups over local fields we compute the etale cohomology of period domains over local fields in the case of a basic isocrystal for quasi split reductive groups period domains which have been introduced by rapoport and zink rz are open admissible rigid analytic subsets of generalized flag varieties they parametrize weakly admissible filtrations of a given isocrystal with additional structure of a reductive group

transonic shock in a nozzle i 2d case in this paper we establish the existence and uniqueness of a transonic shock for the steady flow through a general two dimensional nozzle with variable sections the flow is governed by the inviscid potential equation and is supersonic upstream has no flow boundary conditions on the nozzle walls and an appropriate boundary condition at the exit of the exhaust section the transonic shock is a free boundary dividing two regions of c 1 1 0 flow in the nozzle the potential equation is hyperbolic upstream where the flow is supersonic and elliptic in the downstream subsonic region in particular our results show that there exists a solution to the corresponding free boundary problem such that the equation is always subsonic in the downstream region of the nozzle when the pressure in the exit of the exhaustion section is appropriately larger than that in the entry this problem is motivated by the conjecture of courant and friedrichs on the transonic phenomena in a nozzle 10 furthermore the stability of the transonic shock is also proven when the upstream supersonic flow is a small steady perturbation for the uniform supersonic flow and the corresponding pressure at the exit has a small perturbation the main ingredients of our analysis are a generalized hodograph transformation and multiplier methods for elliptic equation with mixed boundary conditions and comer singularities

uniqueness of transonic shock solutions in a duct for steady potential flow abstract we study the uniqueness of solutions with a transonic shock in a duct in a class of transonic shock solutions which are not necessarily small perturbations of the background solution for steady potential flow we prove that for given uniform supersonic upstream flow in a straight duct there exists a unique uniform pressure at the exit of the duct such that a transonic shock solution exists in the duct which is unique modulo translation for any other given uniform pressure at the exit there exists no transonic shock solution in the duct this is equivalent to establishing a uniqueness theorem for a free boundary problem of a partial differential equation of second order in a bounded or unbounded duct the proof is based on the maximum comparison principle and a judicious choice of special transonic shock solutions as a comparison solution

on the finite volume element method the finite volume element method fve is a discretization technique for partial differential equations it uses a volume integral formulation of the problem with a finite partitioning set of volumes to discretize the equations then restricts the admissible functions to a finite element space to discretize the solution this paper develops discretization error estimates for general selfadjoint elliptic boundary value problems with fve based on triangulations with linear finite element spaces and a general type of control volume we establisho h estimates of the error in a discreteh 1 semi norm under an additional assumption of local uniformity of the triangulation the estimate is improved too h 2 results on the effects of numerical integration are also included

a proof of existence of perturbed steady transonic shocks via a free boundary problem we prove the existence of a solution of a free boundary problem for the transonic small disturbance equation the free boundary is the position of a transonic shock dividing two regions of smooth flow assuming inviscid irrotational flow as modeled by the transonic small disturbance equation the equation is hyperbolic upstream where the flow is supersonic and elliptic in the downstream subsonic region to study the stability of a uniform planar transonic shock we consider perturbation by a steady c1 upstream disturbance if the upstream disturbance is small in a c1 sense then there is a steady solution in which the downstream flow and the transonic shock are holder continuous perturbations of the uniform configuration this result provides a new use of inviscid perturbation techniques to demonstrate in two dimensions the stability of transonic shock waves of the type that appear for example over the wing of an airplane along an airfoil or as bow shocks in a flow with a supersonic free stream velocity 2000 john wiley sons inc

a finite volume method based on the crouzeix raviart element for elliptic pde s in two dimensions we introduce and analyse a finite volume method for the discretization of elliptic boundary value problems in mathbb r 2 the method is based on nonuniform triangulations with piecewise linear nonconforming spaces we prove optimal order error estimates in the l 2 norm and a mesh dependent h 1 norm

transonic shocks for steady euler flows with cylindrical symmetry abstract in this paper we prove the existence of transonic shocks adjacent to a uniform one for the full euler system for steady compressible fluids with cylindrical symmetry in a cylinder and consequently show the stability of such uniform transonic shocks mathematically we solve a free boundary problem for a quasi linear elliptic hyperbolic composite system this reveals that the boundary conditions and equations interact in a subtle way the key point is to separate in a suitable way the elliptic and hyperbolic parts of the system the approach developed here can be applied to deal with certain multidimensional problems concerning stability of transonic shocks for the full euler system

equivariant vector bundles on drinfeld s upper half space let mathcal x subset mathbb p k d be drinfeld s upper half space over a finite extension k of p we construct for every gl d 1 equivariant vector bundle mathcal f on d k a gl d 1 k equivariant filtration by closed subspaces on the k frechet h 0 mathcal x mathcal f this gives rise by duality to a filtration by locally analytic gl d 1 k representations on the strong dual h 0 mathcal x mathcal f prime the graded pieces of this filtration are locally analytic induced representations from locally algebraic ones with respect to maximal parabolic subgroups this paper generalizes the cases of the canonical bundle due to schneider and teitelbaum st1 and that of the structure sheaf by pohlkamp p

multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type we are concerned with the existence and stability of multidimensional transonic shocks for the euler equations for steady potential compressible fluids the euler equations consisting of the conservation law of mass and the bernoulli law for the velocity can be written as the following second order nonlinear equation of mixed elliptic hyperbolic type for the velocity potential r r 1 1 div d d 0 where the density function q is

steady transonic shocks and free boundary problems in infinite cylinders for the euler equations we establish the existence and stability of multidimensional transonic shocks hyperbolic elliptic shocks for the euler equations for steady compressible potential fluids in infinite cylinders the euler equations consisting of the conservation law of mass and the bernoulli law for velocity can be written as a second order nonlinear equation of mixed elliptic hyperbolic type for the velocity potential the transonic shock problem in an infinite cylinder can be formulated into the following free boundary problem the free boundary is the location of the multidimensional transonic shock which divides two regions of c1 flow in the infinite cylinder and the equation is hyperbolic in the upstream region where the c1 perturbed flow is supersonic we develop a nonlinear approach to deal with such a free boundary problem in order to solve the transonic shock problem in unbounded domains our results indicate that there exists a solution of the free boundary problem such that the equation is always elliptic in the unbounded downstream region the uniform velocity state at infinity in the downstream direction is uniquely determined by the given hyperbolic phase and the free boundary is c1 provided that the hyperbolic phase is close in c1 to a uniform flow we further prove that if the steady perturbation of the hyperbolic phase is c2 the free boundary is c2 and stable under the steady perturbation 2003 wiley periodicals inc

an a priori error analysis of the local discontinuous galerkin method for elliptic problems in this paper we present the first a priori error analysis for the local discontinuous galerkin ldg method for a model elliptic problem for arbitrary meshes with hanging nodes and elements of various shapes we show that for stabilization parameters of order one the l2 norm of the gradient and the l2 norm of the potential are of order k and k 1 2 respectively when polynomials of total degree at least k are used if stabilization parameters of order h 1 are taken the order of convergence of the potential increases to k 1 the optimality of these theoretical results is tested in a series of numerical experiments on two dimensional domains

existence of stationary supersonic flows past a pointed body in this paper we study the mathematical aspects of the stationary supersonic flow past a non axisymmetric curved pointed body the flow is described by a steady potential flow equation which is a quasilinear hyperbolic equation of second order we prove the local existence of the solution to this problem with a pointed shock attached at the tip of the pointed body provided the pointed body is a perturbation of a circular cone and the vertex angle of the approximate cone of the pointed body is less than a critical value the solution is smooth in between the shock and the surface of the body consequently such a structure of flow near the tip of the pointed body and its stability is verified mathematically

on the accuracy of the finite volume element method for diffusion equations on composite grids this paper develops the finite volume element fve method which is similar to the so called control volume finite element method but tailored to composite grid applications fve is described for the general two dimensional diffusion equation in divergence form o h 3 2 discretization error estimates are then developed for the case of triangular elements these results are improved to o h 2 using a special modification involving rectilinear elements at the grid interface

semi implicit finite volume scheme for solving nonlinear diffusion equations in image processing we propose and prove a convergence of the semi implicit finite volume approximation scheme for the numerical solution of the modified in the sense of catte lions morel and coll perona malik nonlinear image selective smoothing equation called anisotropic diffusion in the image processing the proof is based on l 2 a priori estimates and kolmogorov s compactness theorem the implementation aspects and computational results are discussed

stability of transonic shock fronts in two dimensional euler systems we study the stability of stationary transonic shock fronts under two dimensional perturbation in gas dynamics the motion of the gas is described by the full euler system the system is hyperbolic ahead of the shock front and is a hyperbolic elliptic composed system behind the shock front the stability of the shock front and the downstream flow under two dimensional perturbation of the upstream flow can be reduced to a free boundary value problem of the hyperbolic elliptic composed system we develop a method to deal with boundary value problems for such systems the crucial point is to decompose the system to a canonical form in which the hyperbolic part and the elliptic part are only weakly coupled in their coefficients by several sophisticated iterative processes we establish the existence and uniqueness of the solution to the described free boundary value problem our result indicates the stability of the transonic shock front and the flow field behind the shock

on the accuracy of the finite volume element method based on piecewise linear polynomials we present a general error estimation framework for a finite volume element fve method based on linear polynomials for solving second order elliptic boundary value problems this framework treats the fve method as a perturbation of the galerkin finite element method and reveals that regularities in both the exact solution and the source term can affect the accuracy of fve methods in particular the error estimates and counterexamples in this paper will confirm that the fve method cannot have the standard o h2 convergence rate in the l2 norm when the source term has the minimum regularity only being in l2 even if the exact solution is in h2

stability of spherically symmetric subsonic flows and transonic shocks under multidimensional perturbations abstract we develop a method that works in general product riemannian manifold to decompose the three dimensional steady full compressible euler system which is of elliptic hyperbolic composite mixed type for subsonic flows the method is applied to show stability of spherically symmetric subsonic flows and transonic shocks in space r 3 under multidimensional perturbations of boundary conditions

on special representations of p adic reductive groups let f be a non archimedean locally compact field let g be a split connected reductive group over f for a parabolic subgroup q g and a ring l we consider the g representation on the l module c g q l

compressible flow and transonic shock in a diverging nozzle the paper is devoted to the study of compressible flows and transonic shocks in diverging nozzles in the framework of the full compressible euler system consider a nozzle having a shape as a diverging truncated sector with generic opening angle if the upstream flow at the entrance is supersonic and is near to an axial symmetric flow and if all parameters of the upstream flow and the receiver pressure at the exit are suitably assigned then a transonic shock appears in the nozzle to determine the transonic shock and the flow in the nozzle leads to a free boundary value problem for a nonlinear partial differential equation we prove that the receiver pressure can uniquely determine the location of the transonic shock as well as the flow behind the shock such a conclusion was conjectured by courant and friedrichs and is confirmed theoretically in this paper for the divergent nozzles

on the construction and analysis of high order locally conservative finite volume type methods for one dimensional elliptic problems locally conservative finite volume type methods based on continuous piecewise polynomial functions of degree r ge 2 are introduced and analyzed in the context of indefinite elliptic problems in one space dimension the new methods extend and generalize the classical finite volume method based on piecewise linear functions we derive a priori error estimates in the l2 h1 and l infty norm and discuss superconvergence effects for the error and its derivative explicit residual based a posteriori error bounds in the l2 and energy norm are also derived we compute the experimental order of convergence and show the results of an adaptive algorithm based on the a posteriori error estimates

a posteriori error estimation and adaptive mesh refinement techniques abstract we analyse three different a posteriori error estimators for elliptic partial differential equations they are based on the evaluation of local residuals with respect to the strong form of the differential equation on the solution of local problems with neumann boundary conditions and on the solution of local problems with dirichlet boundary conditions we prove that all three are equivalent and yield global upper and local lower bounds for the true error thus adaptive mesh refinement techniques based on these estimators are capable to detect local singularities of the solution and to appropriately refine the grid near these singularities some numerical examples prove the efficiency of the error estimators and the mesh refinement techniques

transonic shocks in compressible flow passing a duct for three dimensional euler systems in this paper we study the transonic shock in steady compressible flow passing a duct the flow is a given supersonic one at the entrance of the duct and becomes subsonic across a shock front which passes through a given point on the wall of the duct the flow is governed by the three dimensional steady full euler system which is purely hyperbolic ahead of the shock and is of elliptic hyperbolic composed type behind the shock the upstream flow is a uniform supersonic one with the addition of a three dimensional perturbation while the pressure of the downstream flow at the exit of the duct is assigned apart from a constant difference the problem of determining the transonic shock and the flow behind the shock is reduced to a free boundary value problem in order to solve the free boundary problem of the elliptic hyperbolic system one crucial point is to decompose the whole system to a canonical form in which the elliptic part and the hyperbolic part are separated at the level of the principal part due to the complexity of the characteristic varieties for the three dimensional euler system the calculus of symbols is employed to complete the decomposition the new ingredient of our analysis also contains the process of determining the shock front governed by a pair of partial differential equations which are coupled with the three dimensional euler system