a semismooth equation approach to the solution of nonlinear complementarity problems in this paper we present a new algorithm for the solution of nonlinear complementarity problems the algorithm is based on a semismooth equation reformulation of the complementarity problem we exploit the recent extension of newton s method to semismooth systems of equations and the fact that the natural merit function associated to the equation reformulation is continuously differentiable to develop an algorithm whose global and quadratic convergence properties can be established under very mild assumptions other interesting features of the new algorithm are an extreme simplicity along with a low computational burden per iteration we include numerical tests which show the viability of the approach

on h p adaptive bem for frictional contact problems in linear elasticity a mixed formulation for a tresca frictional contact problem in linear elasticity is considered in the context of boundary integral equations which is later extended to coulomb friction the discrete lagrange multiplier an approximation of the surface traction on the contact boundary part is a linear combination of biorthogonal basis functions in case of curved isoparametric elements these are the solutions of local problems in particular the biorthogonality allows to rewrite the variational inequality constraints as a simple set of complementarity problems thus enabling an efficient application of a semi smooth newton solver for the discrete mixed problem converging locally super linearly in the frictional case and quadratically in the frictionless case typically the solution of frictional contact problems is of reduced regularity at the interface between contact to non contact and from stick to slip to identify the a priori unknown locations of these interfaces two a posteriori error estimations are introduced in a first step the error is split into specific error contributions resulting from the contact and friction conditions and from the discretization error of a variational equation for the latter a residual and a bubble error estimation are considered explicitly the numerical experiments show the applicability of the derived error estimations and the superiority of h p adaptivity compared to low order uniform and adaptive approaches

remarks on codes from hermitian curves corresp parameters and generator matrices are given for the codes obtained by applying goppa s algebraic geometric construction method to hermitian curves in pg 2 q where q 2 2s for some s in bf n automorphisms of these codes are also discussed and some results on self duality and weak self duality are given

comparison of mixed h p bem stabilized and non stabilized for frictional contact problems we present boundary integral equation procedures for contact problems with tresca or coulomb friction which are based on mixed formulations where besides the displacement also the traction on the contact boundary part appears as a lagrange multiplier this approach allows for an easy and efficient way to perform an h p be method by the use of biorthogonal basis functions this is especially suited for applying the semi smooth newton ssn method which is a very efficient solver superior to standard algorithm like uzawa with an adaptive algorithm we perform locally mesh refinements and increase of polynomial degrees for the be solution thus correctly representing the contact phenomena we also present as stabilized version of our mixed h p bem scheme with gauss lobatto lagrange basis which circumvents the discrete inf sup condition numerical results supporting our theory are reported

polynomial interpolation results in sobolev spaces abstract in this paper we derive new results related to the lagrange polynomial interpolation on the nodes of gauss type quadrature formulas in particular we obtain asymptotic error estimates of the same order as the distance to the best fit in the polynomial space

semismooth karush kuhn tucker equations and convergence analysis of newton and quasi newton methods for solving these equations there are several forms of systems of nonsmooth equations which are equivalent to the karush kuhn tucker kkt system of a nonlinearly constrained optimization problem nlp if the nlp is twice continuously differentiable and the hessian functions of its objective and constraint functions are locally lipschitzian then these kkt equations are strongly semismooth if furthermore the linear independence condition and the strong second order sufficiency condition are satisfied at a kkt point then the generalized jacobians of these kkt equations are nonsingular at that point and the sequence generated by the generalized newton method converges to this point q quadratically however direct application of quasi newton methods cannot guarantee q superlinear convergence we present a mixed quasi newton method which converges q superlinearly with common symmetrical updating rules under the above conditions for the generalized newton method superlinear convergence of the primal variables and global convergence are also discussed

torsion limits and riemann roch systems for function fields and applications the ihara limit or constant a q has been a central problem of study in the asymptotic theory of global function fields or equivalently algebraic curves over finite fields it addresses global function fields with many rational points and so far most applications of this theory do not require additional properties motivated by recent applications we require global function fields with the additional property that their zero class divisor groups contain at most a small number of d torsion points we capture this with the notion of torsion limit a new asymptotic quantity for global function fields it seems that it is even harder to determine values of this new quantity than the ihara constant nevertheless some nontrivial upper bounds are derived apart from this new asymptotic quantity and bounds on it we also introduce riemann roch systems of equations it turns out that this type of equation system plays an important role in the study of several other problems in each of these areas arithmetic secret sharing symmetric bilinear complexity of multiplication in finite fields frameproof codes and the theory of error correcting codes finally we show how our new asymptotic quantity our bounds on it and riemann roch systems can be used to improve results in these areas

a note on the design of hp adaptive finite element methods for elliptic partial differential equations we introduce an hp adaptive finite element algorithm based on a combination of reliable and efficient residual error indicators and a new hp extension control technique which assesses the local regularity of the underlying analytical solution on the basis of its local legendre series expansion numerical experiments confirm the robustness and reliability of the proposed algorithm

direct calculation of accurate siegert eigenvalues the roots of the hankel determinant built with the taylor coefficients of the regularized logarithmic derivative of the eigenfunction give both the bound state and siegert eigenvalues resonances of separable quantum mechanical systems we obtain accurate positions and widths for the resonances of some commonly chosen test examples

solving linear variational inequality problems by a self adaptive projection method we propose a self adaptive projection method for solving linear variational inequality problems and show its global convergence under mild conditions some numerical results are also addressed which indicate that the method is quite robust and efficient

explicit construction of ag codes from a curve in the tower of bassa beelen garcia stichtenoth we present multi point algebraic geometric codes overstepping the gilbert varshamov bound the construction is based on a curve in the tower introduced by bassa et al these codes are described in detail by constructing generator matrices it turns out that these codes have nice properties similar to those of hermitian codes the duals are also such codes and can be expressed as an explicit formula

on the summation of some divergent hypergeometric series and related perturbation expansions abstract divergent hypergeometric series 2 f 0 1 occur frequently in poincare type asymptotic expansions of special functions these divergent series 2 f 0 can be used for the evaluation of the corresponding special functions if suitable summation technique are applied there is considerable evidence that levin s sequence transformation 1973 and in particular also some closely related sequence transformations weniger 1989 which were derived recently sum divergent series 2 f 0 much more efficiently than pade approximants similar summation problems occur also in the case of divergent rayleigh schrodinger perturbation expansions of elementary quantum mechanical systems a comparison of the perturbation series for the quartic anharmonic oscillator with the closely related asymptotic series for the complementary error function shows that levin s sequence transformation and the recently derived new sequence transformations are again more efficient than pade approximants however the superiority of levin s sequence transformation and of the new sequence transformations is less pronounced in the case of the perturbation expansion

on the μx2 λx4 ηx6 interaction a class of exact even and odd parity solutions of the schrodinger equation for the interaction x2 x4 x6 0 is obtained in the form polynomial exponential when and satisfy some specific relations in the general case the eigenvalues of the bounded potential problem are found by the method of series solution for any value of the authors also show that the analytic continued fraction method does not always lead to the corrected results

the h p version of the coupling of finite element and boundary element methods for transmission problems in polyhedral domains this paper analyzes the rate of convergence of the h p version of the coupling of the finite element and boundary element method for transmission problems with a linear differential operator with variable coefficients in a bounded polyhedral domain omega 1 and with constant coefficients in the exterior domain omega 2 bbb r 3 backslash overline omega 1 this procedure uses the variational formulation of the differential equation in omega 1 and involves integral operators on the interface between omega 1 and omega 2 the finite elements are used to obtain approximate solutions of the differential equation in omega 1 and the boundary elements are used to obtain approximate solutions of the integral equations for given piecewise analytic data we show that the galerkin solution of this coupling procedure converges exponentially fast in the energy norm if the h p version is used both for finite elements and boundary elements

an adaptive wavelet collocation method for solving optimal control of elliptic variational inequalities of the obstacle type abstract this paper presents a fast computational technique based on the wavelet collocation method for the numerical solution of an optimal control problem governed by elliptic variational inequalities of obstacle type in this problem the solution divides the domain into contact and noncontact sets the boundary between the contact and noncontact sets is a free boundary which is not known a priori and the solution is not smooth on it accordingly a very fine grid is needed in order to obtain a solution with a reasonable accuracy in this paper our aim is to propose an adaptive scheme in order to generate an appropriate and economic irregular dyadic mesh for finding the optimal control and state functions the irregular mesh will be generated such that its density around the free boundary is higher than in other places and high resolution computations are focused on these zones to this aim we use an adaptive wavelet collocation method and take advantage of the fast wavelet transform of compact supported interpolating wavelets to develop a multi level algorithm which generates an adaptive computational grid using this adaptive grid takes less cpu time than using a full regular mesh at each step of the algorithm the active set method is used for solving the optimality system of the obstacle problem on the adapted mesh finally the numerical examples are presented to show the validity and efficiency of the technique

perturbative solution for the generalised anharmonic oscillators serious difficulties have been encountered in the past when attempting the straightforward application of the borel method to the summation of perturbation series in quantum mechanics and quantum field theory because of an analytic continuation of the borel transform series which is usually performed by means of conformal mapping techniques the author shows that this continuation may be accomplished by combining the borel method with the first confluent form of the epsilon algorithm of wynn 1960 1968 for the calculation of the ground state energy eigenvalue of the generalised anharmonic oscillators two different approaches are presented both of which lead to accurate results

small data oscillation implies the saturation assumption the saturation assumption asserts that the best approximation error in h 1 0 with piecewise quadratic finite elements is strictly smaller than that of piecewise linear finite elements we establish a link between this assumption and the oscillation of f delta u and prove that small oscillation relative to the best error with piecewise linears implies the saturation assumption we also show that this condition is necessary and asymptotically valid provided f in l 2

the eigenvalues of the bounded λx2m oscillators the even and odd parity eigenvalues of the bounded pure x2m oscillators are obtained as the roots of two equations derived explicitly

the anharmonic oscillator perturbation series for cubic and quartic energy distortion twenty five terms of the perturbation series are calculated for the first five energy levels of an oscillator with lambda x4 energy distortion and a similar calculation is carried out for the first six levels for lambda x3 distortion for the positive quartic the alternating series is summed using the aitken delta 2 transformation for lambda from 0 1 to 100 an exponential integral model and a truncated binomial model are used to transform the series for negative distortion energies and the complex energy eigenvalues are calculated for lambda from 0 01 to 1

biorthogonal bases with local support and approximation properties we construct locally supported basis functions which are biorthogonal to conforming nodal finite element basis functions of degree p in one dimension in contrast to earlier approaches these basis functions have the same support as the nodal finite element basis functions and reproduce the conforming finite element space of degree p 1 working with gaus lobatto nodes we find an interesting connection between biorthogonality and quadrature formulas one important application of these newly constructed biorthogonal basis functions are two dimensional mortar finite elements the weak continuity condition of the constrained mortar space is realized in terms of our new dual bases as a result local static condensation can be applied which is very attractive from the numerical point of view numerical results are presented for cubic mortar finite elements

generalized hermitian codes over hboxgf 2 r in this correspondence a generalization of hermitian function field proposed by garcia and stichtenoth is studied a weierstrass semigroup of the point at infinity for the case q 2 r ges 3 is calculated it turns out that unlike for the hermitian case there are already three generators for the semigroup this result then is applied to codes constructed on generalized hermitian gh function fields further results of kirfel and pellikaan are applied to estimating a feng rao designed distance for gh codes which improve on the goppa designed minimum distance next the question of codes dual to gh codes is studied it is shown that the duals are also gh codes and an explicit formula is given in particular this formula enables one to calculate the parameters of a dual code a new record giving 32 16 ges 12 code over gf 8 is presented as one of the examples

a posteriori error estimates of discontinuous galerkin methods for the signorini problem a reliable and efficient a posteriori error estimator is derived for a class of discontinuous galerkin dg methods for the signorini problem a common property shared by many dg methods leads to a unified error analysis with the help of a constraint preserving enriching map the error estimator of dg methods is comparable with the error estimator of the conforming methods numerical experiments illustrate the performance of the error estimator

option pricing under a double exponential jump diffusion model this paper aims to extend the analytical tractability of the black scholes model to alternative models with arbitrary jump size distributions more precisely we propose a jump diffusion model for asset prices whose jump sizes have a mixed exponential distribution which is a weighted average of exponential distributions but with possibly negative weights the new model extends existing models such as hyperexponential and double exponential jump diffusion models because the mixed exponential distribution can approximate any distribution as closely as possible including the normal distribution and various heavy tailed distributions the mixed exponential jump diffusion model can lead to analytical solutions for laplace transforms of prices and sensitivity parameters for path dependent options such as lookback and barrier options the laplace transforms can be inverted via the euler inversion algorithm numerical experiments indicate that the formulae are easy to implement and accurate the analytical solutions are made possible mainly because we solve a high order integro differential equation explicitly a calibration example for spy options shows that the model can provide a reasonable fit even for options with very short maturity such as one day this paper was accepted by michael fu stochastic models and simulation

multi point codes from generalized hermitian curves we investigate multi point algebraic geometric codes defined from curves related to the generalized hermitian curve introduced by bassa et al our main result is to find a basis of the riemann roch space of a series of divisors which can be used to construct multi point codes explicitly these codes turn out to have nice properties similar to those of hermitian codes for example they are easy to describe to encode and decode it is shown that the duals are also such codes and an explicit formula is given in particular this formula enables one to calculate the parameters of these codes finally we apply our results to obtain linear codes attaining new records on the parameters a new record giving 234 141 59 code over mathbb f 27 is presented as one of the examples

mixed finite element methods of higher order for model contact problems this paper presents mixed finite element methods of higher order for a simplified signorini problem and an idealized frictional problem the discretization is based on a mixed variational formulation proposed by haslinger et al which is extended to higher order finite elements to guarantee the unique existence of the solution of the mixed method a discrete inf sup condition is proven approximation results of the p method of finite elements and some inverse estimates for higher order polynomials are applied numerical results confirm the theoretical findings

a priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems we analyze three discontinuous galerkin approximations for solving elliptic problems in two or three dimensions in each one the basic bilinear form is nonsymmetric the first one has a penalty term on edges the second has one constraint per edge and the third is totally unconstrained for each of them we prove hp error estimates in the h1 norm optimal with respect to h the mesh size and nearly optimal with respect to p the degree of polynomial approximation we establish these results for general elements in two and three dimensions for the unconstrained method we establish a new approximation result valid on simplicial elements l2 estimates are also derived for the three methods

conforming and discontinuous galerkin fem in space for solving parabolic obstacle problem abstract in this article we propose and analyze conforming and discontinuous galerkin dg finite element methods for numerical approximation of the solution of the parabolic variational inequality associated with a general obstacle in r d d 2 3 for fully discrete conforming method we use globally continuous and piecewise linear finite element space whereas for the fully discrete dg scheme we employ piecewise linear finite element space for spatial discretization the time discretization has been done by using the implicit backward euler method we present the error analysis for the conforming and the dg fully discrete schemes and derive an error estimate of optimal order o h t in a certain energy norm defined precisely in the article the analysis is performed without any assumptions on the speed of propagation of the free boundary but only assumes the pragmatic regularity that u t l 2 0 t l 2 the obstacle constraints are incorporated at the lagrange nodes of the triangular mesh and the analysis exploits the lagrange interpolation we present some numerical experiment to illustrate the performance of the proposed methods

hp adaptive ipdg tdg fem for parabolic obstacle problems for a parabolic obstacle problem two equivalent hp fem discretization methods based on interior penalty discontinuous galerkin in space and discontinuous galerkin in time are presented the first approach is based on a variational inequality vi formulation and the second approach on a mixed method in which the non penetration condition is resolved by a lagrange multiplier the discrete lagrange multiplier is a linear combination of biorthogonal basis functions allowing to write the discrete vi constraints as a set of complementarity problems employing a penalized fischer burmeister non linear complementarity function the discrete mixed problem can be solved by a locally q quadratic converging semi smooth newton ssn method the hierarchical a posteriori error estimator for the vi formulation which under the saturation assumption is both efficient and reliable allows hp adaptivity the numerical experiments show improved convergence compared to uniform and h adaptive meshes furthermore an a priori error estimate is given for the vi formulation

time discretization of parabolic problems by the hp version of the discontinuous galerkin finite element method the discontinuous galerkin finite element method dgfem for the time discretization of parabolic problems is analyzed in the context of the hp version of the galerkin method error bounds which are explicit in the time steps as well as in the approximation orders are derived and it is shown that the hp dgfem gives spectral convergence in problems with smooth time dependence in conjunction with geometric time partitions it is proved that the hp dgfem results in exponential rates of convergence for piecewise analytic solutions exhibiting singularities induced by incompatible initial data or piecewise analytic forcing terms for the h version dgfem algebraically graded time partitions are determined that give the optimal algebraic convergence rates a fully discrete hp scheme is discussed exemplarily for the heat equation the use of certain mesh design principles for the spatial discretizations yields exponential rates of convergence in time and space numerical examples confirm the theoretical results

biorthogonal basis functions in hp adaptive fem for elliptic obstacle problems abstract in this paper the discretization of a non symmetric elliptic obstacle problem with h p adaptive h 1 conforming finite elements is discussed for this purpose a higher order mixed finite element discretization is introduced where the dual space is discretized via biorthogonal basis functions the h p adaptivity is realized via automatic adaptive mesh refinement based on a residual a posteriori error estimation which is also derived in this paper the use of biorthogonal basis functions leads to unilateral box constraints and componentwise complementarity conditions enabling the highly efficient application of a quadratically converging semi smooth newton scheme which can be modified to ensure global convergence h p adaptivity usually implies meshes with hanging nodes and varying polynomial degrees which have to be handled appropriately within the h 1 conforming finite element discretization this is typically done by using so called connectivity matrices in this paper a procedure is proposed which efficiently computes these matrices for biorthogonal basis functions finally the applicability of the theoretical findings is demonstrated with several numerical experiments

summation of asymptotic expansions of multiple valued functions using algebraic approximants application to anharmonic oscillators the divergent rayleigh schrodinger perturbation expansions for energy eigenvalues of cubic quartic sextic and octic oscillators are summed using algebraic approximants these approximants are generalized pade approximants that are obtained from an algebraic equation of arbitrary degree numerical results indicate that given enough terms in the asymptotic expansion the rate of convergence of the diagonal staircase approximant sequence increases with the degree different branches of the approximants converge to different branches of the function the success of the high degree approximants is attributed to their ability to model the function on multiple sheets of the riemann surface and to reproduce the correct singularity structure in the limit of large perturbation parameter an efficient recursive algorithm for computing the diagonal approximant sequence is presented

the anharmonic oscillator complex eigenvalues for the ground state with negative quartic or cubic energy distortion the perturbed ground state eigenvalues for the harmonic oscillator with negative quartic and with cubic energy distortion are calculated by numerical integration these agree with the author s results using an asymptotic series

on the weight hierarchy of geometric goppa codes the weight hierarchy of a linear code is the set of generalized hamming weights of the code in the paper the authors consider geometric goppa codes and provide a lower bound on their generalized hamming weights similar to goppa s lower bound on their minimum distance in the particular case of hermitian codes exact results on the second and third generalized hamming weights are given for any m except a few cases where m is a parameter that governs the dimension of these codes in many instances the authors are able to provide considerably more information on their generalized hamming weights an upper bound relating the generalized hamming weights of hermitian codes to the pole numbers at a special point on the curve is also provided similar results are given in the case of codes from some subfields of the hermitian function fields which are also maximal finally a nontrivial family of codes is also presented whose weight hierarchy is completely determined

complex variable hvpt complex variable hypervirial perturbation theory is applied to the case of oscillator and coulomb potentials perturbed by a single term potential of the form vxn or vrn respectively the trial calculations reported show that this approach can produce accurate complex energies for resonant states via a simple and speedy calculation and can also be useful in studies of pt symmetry and tunnelling resonance effects

an infeasible active set method for quadratic problems with simple bounds a primal dual active set method for quadratic problems with bound constraints is presented based on a guess on the active set a primal dual pair x s is computed that satisfies the first order optimality condition and the complementarity condition if x s is not feasible a new active set is determined and the process is iterated sufficient conditions for the iterations to stop in a finite number of steps with an optimal solution are provided computational experience indicates that this approach often requires only a few less than 10 iterations to find the optimal solution

a mortar finite element method using dual spaces for the lagrange multiplier the mortar finite element method allows the coupling of different discretization schemes and triangulations across subregion boundaries in the original mortar approach the matching at the interface is realized by enforcing an orthogonality relation between the jump and a modified trace space which serves as a space of lagrange multipliers in this paper this lagrange multiplier space is replaced by a dual space without losing the optimality of the method the advantage of this new approach is that the matching condition is much easier to realize in particular all the basis functions of the new method are supported in a few elements the mortar map can be represented by a diagonal matrix in the standard mortar method a linear system of equations must be solved the problem is considered in a positive definite nonconforming variational as well as an equivalent saddle point formulation

the dual minimum distance of arbitrary dimensional algebraic geometric codes abstract in this article the minimum distance of the dual c of a functional code c on an arbitrary dimensional variety x over a finite field f q is studied the approach is based on problems a la cayley bacharach and consists in describing the minimal configurations of points on x which fail to impose independent conditions on forms of some degree m if x is a curve the result improves in some situations the well known goppa designed distance

a posteriori error estimation for hp version time stepping methods for parabolic partial differential equations the aim of this paper is to develop an hp version a posteriori error analysis for the time discretization of parabolic problems by the continuous galerkin cg and the discontinuous galerkin dg time stepping methods respectively the resulting error estimators are fully explicit with respect to the local time steps and approximation orders their performance within an hp adaptive refinement procedure is illustrated with a series of numerical experiments

a posteriori error estimates of higher order finite elements for frictional contact problems abstract in this paper a posteriori estimates are derived for higher order finite element methods and frictional contact problems the discretization is based on a mixed approach where the geometrical and frictional constraints are captured by lagrange multipliers the use of higher order polynomials leads to a certain non conformity in the discretization which requires special attention in the error analysis as a main result an error estimation is proposed which consists of the dual norm of a residual plus some computable remainder terms the residual is estimated by well known a posteriori error estimates for variational equations the remainder terms represent typical sources resulting from the non conforming mixed discretization numerical experiments confirm the applicability of the a posteriori estimates to adaptive mesh refinements

a p version finite element method for nonlinear elliptic variational inequalities in 2d this article introduces and analyzes a p version fem for variational inequalities resulting from obstacle problems for some quasi linear elliptic partial differential operators we approximate the solution by controlling the obstacle condition in images of the gauss lobatto points we show existence and uniqueness for the discrete solution up from the p version for the obstacle problem we prove the convergence of up towards the solution with respect to the energy norm and assuming some additional regularity for the solution we derive an a priori error estimate in numerical experiments the p version turns out to be superior to the h version concerning the convergence rate and the number of unknowns needed to achieve a certain exactness of the approximation

a penalized fischer burmeister ncp function we introduce a new ncp function in order to reformulate the nonlinear complementarity problem as a nonsmooth system of equations this new ncp function turns out to have stronger theoretical properties than the widely used fischer burmeister function and other ncp functions suggested previously moreover numerical experience indicates that a semismooth newton method based on this new ncp function performs considerably better than the corresponding method based on the fischer burmeister function

on the x2 x4 interaction the even and odd parity eigenvalues for the bounded potential mu x2 lambda x4 with both positive and negative values of mu and lambda 0 are obtained by the method of series solution

a primal dual active set strategy for non linear multibody contact problems abstract non conforming domain decomposition methods provide a powerful tool for the numerical approximation of partial differential equations for the discretization of a non linear multibody contact problem we use the mortar approach with a dual lagrange multiplier space to handle the non linearity of the contact conditions we apply a primal dual active set strategy to find the actual contact zone the algorithm can be easily generalized to multibody contact problems a suitable basis transformation guarantees the same algebraic structure in the multibody situation as in the one body case using an inexact primal dual active set strategy in combination with a multigrid method yields an efficient iterative solver different numerical examples for one and multibody contact problems illustrate the performance of the method

quantum anharmonic symmetrical oscillators using elliptic functions the authors study in the jwkb approximation the energy levels of the symmetric anharmonic oscillators v x ax2 bx4 for different signs and values of a and b comparisons are made with published results for specific cases and with numerical calculations they given an additional example of exact value to add to the very rare catalogue of known examples

space time adaptive finite elements for nonlocal parabolic variational inequalities abstract this article considers the error analysis of finite element discretizations and adaptive mesh refinement procedures for nonlocal dynamic contact and friction both in the domain and on the boundary for a large class of parabolic variational inequalities associated to the fractional laplacian we obtain a priori and a posteriori error estimates and study the resulting space time adaptive mesh refinement procedures particular emphasis is placed on mixed formulations which include the contact forces as a lagrange multiplier corresponding results are presented for elliptic problems our numerical experiments for 2 dimensional model problems confirm the theoretical results they indicate the efficiency of the a posteriori error estimates and illustrate the convergence properties of space time adaptive as well as uniform and graded discretizations