on the optimal shape parameter for gaussian radial basis function finite difference approximation of the poisson equation we investigate the influence of the shape parameter in the meshless gaussian radial basis function finite difference rbf fd method with irregular centres on the quality of the approximation of the dirichlet problem for the poisson equation with smooth solution numerical experiments show that the optimal shape parameter strongly depends on the problem but insignificantly on the density of the centres therefore we suggest a multilevel algorithm that effectively finds a near optimal shape parameter which helps to significantly reduce the error comparison to the finite element method and to the generalised finite differences obtained in the flat limits of the gaussian rbf is provided

a collection of 2d elliptic problems for testing adaptive grid refinement algorithms adaptive grid refinement is a critical component of the improvements that have recently been made in algorithms for the numerical solution of partial differential equations pdes the development of new algorithms and computer codes for the solution of pdes usually involves the use of proof of concept test problems 2d elliptic problems are often used as the first test bed for new algorithms and codes this paper contains a set of twelve parametrized 2d elliptic test problems for adaptive grid refinement algorithms and codes the problems exhibit a variety of types of singularities near singularities and other difficulties

modules without self extensions over gentle algebras abstract let x be a module over a gentle algebra such that ext 1 x x 0 then end x is up to morita equivalence a gentle algebra too

blended ausm method for all speeds and all grid aspect ratios an ausm based discretization method using an explicit third order discretization for the convective part and a line implicit central discretization for the acoustic part and for the diffusive part has been developed for incompressible and low speed compressible navier stokes equations the lines are chosen in the direction of the gridpoints with shortest connection the semi implicit line method is used in multistage form because of the explicit third order discretization of the convective part multigrid is used as acceleration technique because of the implicit treatment of the acoustic and the diffusive terms the stiffness otherwise caused by high aspect ratio cells is removed low mach number stiffness is treated by a preconditioning technique to ensure physical correct behavior of the discretization for vanishing mach number extreme care has been taken for vanishing mach number stabilization terms are added to the mass flux pressure and temperature stabilization terms are necessary the coefficients of these terms are chosen so that correct scaling with mach number is obtained a blend of the low speed algorithm with the original ausm algorithm developed for high speed applications has been constructed so that the resulting algorithm can be used at all speeds

adaptive rbf fd method for elliptic problems with point singularities in 2d abstract we describe and test numerically an adaptive meshless generalized finite difference method based on radial basis functions that competes well with the finite element method on standard benchmark problems with reentrant corners of the boundary sharp peaks and rapid oscillations in the neighborhood of an isolated point this is achieved thanks to significant improvements introduced into the earlier algorithms of davydov and oanh 2011 including a new error indicator of zienkiewicz zhu type

a simple error estimator and adaptive procedure for practical engineerng analysis a new error estimator is presented which is not only reasonably accurate but whose evaluation is computationally so simple that it can be readily implemented in existing finite element codes the estimator allows the global energy norm error to be well estmated and alos gives a good evaluation of local errors it can thus be combined with a full adaptive process of refinement or more simply provide guidance for mesh redesign which allows the user to obtain a desired accuracy with one or two trials when combined with an automatic mesh generator a very efficient guidance process to analysis is avaiable estimates other than the energy norm have successfully been applied giving for instance a predetermined accuracy of stresses

maps between tree and band modules the indecomposable modules over special biserial algebras are known to be of a specially simple form so called tree and band modules cf 6 2 crawley boevey has shown how to describe the homomorphisms between tree modules cf 3 and our objective in this paper is to extend this result to band modules tree and band modules occur as indecomposable representations of arbitrary algebras nevertheless we follow crawley boevey and consider zero relation algebras as the appropriate context given two tree or band modules we first study certain quiver homomorphisms between their underlying trees or bands and it turns out that a map between the representations is completely described by k linear maps which are associated to these quiver homomorphisms it is interesting to note that in general the classification of maps between representations of a special biserial algebra a is a wild problem in fact the maps are equivalent to the representations of t a for instance take for a the path algebra of an a n 2 6 then t a is wild although the maps between indecomposable representations of a are easily to describe throughout k will be a fixed algebraically closed field maps are written on the left this paper results from a stay at the university of liverpool supported by the deutscher akademischer austauschdienst i am very grateful to sheila brenner for her suggestions

simultaneous solution algorithms for eulerian eulerian gas solid flow models stability analysis and convergence behaviour of a point and a plane solver simultaneous solution algorithms for eulerian eulerian gas solid flow models are presented and their stability analyzed the integration algorithms are based on dual time stepping with fourth order runge kutta in pseudo time the domain is solved point or plane wise the discretization of the inviscid terms is based on a low mach limit of the multi phase preconditioned advection upstream splitting method mp ausmp the numerical stability of the simultaneous solution algorithms is analyzed in 2d with the fourier method stability results are compared with the convergence behaviour of 3d riser simulations the impact of the grid aspect ratio preconditioning artificial dissipation and the treatment of the source terms is investigated a particular advantage of the simultaneous solution algorithms is that they allow a fully implicit treatment of the source terms which are of crucial importance for the eulerian eulerian gas solid flow models and their solution the numerical stability of the optimal simultaneous solution algorithm is analyzed for different solids volume fractions and gas solid slip velocities furthermore the effect of the grid resolution on the convergence behaviour and the simulation results is investigated finally simulations of the bottom zone of a pilot scale riser with a side solids inlet are experimentally validated

a two stage adaptive scheme based on rbf collocation for solving elliptic pdes abstract in this paper we present a new adaptive two stage algorithm for solving elliptic partial differential equations via a radial basis function collocation method our adaptive meshless scheme is based at first on the use of a leave one out cross validation technique and then on a residual subsampling method each of phases is characterized by different error indicators and refinement strategies the combination of these computational approaches allows us to detect the areas that need to be refined also including the chance to further add or remove adaptively any points the resulting algorithm turns out to be flexible and effective through a good interaction between error indicators and refinement procedures several numerical experiments support our study by illustrating the performance of our two stage scheme finally the latter is also compared with an efficient adaptive finite element method

error bounds for kernel based numerical differentiation the literature on meshless methods shows that kernel based numerical differentiation formulae are robust and provide high accuracy at low cost this paper analyzes the error of such formulas using the new technique of growth functions it allows to bypass certain technical assumptions that were needed to prove the standard error bounds on interpolants and their derivatives since differentiation formulas based on polynomials also have error bounds in terms of growth functions we have a convenient way to compare kernel based and polynomial based formulas it follows that kernel based formulas are comparable in accuracy to the best possible polynomial based formulas a variety of examples is provided

exceptional modules are tree modules abstract we consider representations of a quiver over an arbitrary field recall that an indecomposable representation m without self extensions is said to be exceptional we are going to show that exceptional representations can be exhibited using matrices involving as coefficients just 0 and 1 actually if d is the dimension of m there exists such a matrix presentation with precisely d 1 non zero coefficients the corresponding coefficient quiver is a tree

a multigrid semi implicit line method for viscous incompressible and low mach number flows on high aspect ratio grids abstract discretization of viscous incompressible and viscous low mach number flows often leads to a system of equations which is very difficult to solve there are two reasons first the use of high aspect ratio grids results in a very numerically anisotropic behaviour of the diffusive and acoustic terms and second in low mach number flow the ratio of the convective and acoustic eigenvalues of the inviscid system becomes very high we implemented an ausm based discretization method using an explicit third order discretization for the convective part and a line implicit central discretization for the acoustic part and for the diffusive part the lines are chosen in the direction of the grid points with shortest connection the preconditioned semi implicit line method is used in multistage form because of the explicit third order discretization of the convective part multigrid is used as an acceleration technique it is shown that the convergence is very good independent of grid aspect ratio and mach number

development of low diffusion flux splitting methods for dense gas solid flows the development of a class of low diffusion upwinding methods for computing dense gas solid flows is presented in this work an artificial compressibility low mach preconditioning strategy is developed for a hyperbolic two phase flow equation system consisting of separate solids and gas momentum and continuity equations the eigenvalues of this system are used to devise extensions of the ausm 1 and ldfss 2 flux splitting methods that provide high resolution capturing of bubble growth and collapse in gas solid fluidized beds applications to several problems in fluidization are presented

preconditioned methods for solving the incompressible low speed compressible equations abstract acceleration methods are presented for solving the steady state incompressible equations these systems are preconditioned by introducing artificial time derivatives which allow for a faster convergence to the steady state we also consider the compressible equations in conservation form with slow flow two arbitrary functions and are introduced in the general preconditioning an analysis of this system is presented and an optimal value for is determined given a constant it is further sown that the resultant incompressible equations form a symmetric hyperbolic system and so are well posed several generalizations to the compressible equations are presented which extend previous results

krull gabriel dimension of domestic string algebras we classify the indecomposable pure injective modules over a wide class of 1 domestic string algebras and calculate the krull gabriel dimension of these algebras

cotorsion pairs in the cluster category of a marked surface abstract we study extension spaces cotorsion pairs and their mutations in the cluster category of a marked surface without punctures under the one to one correspondence between the curves valued closed curves in the marked surface and the indecomposable objects in the associated cluster category we prove that the dimension of extension space of two indecomposable objects in the cluster categories equals to the intersection number of the corresponding curves by using this result we prove that there are no non trivial t structures in the cluster categories when the surface is connected based on this result we give a classification of cotorsion pairs in these categories moreover we define the notion of paintings of a marked surface without punctures and their rotations they are a geometric model of cotorsion pairs and of their mutations respectively

a simple mesh generator in matlab creating a mesh is the first step in a wide range of applications including scientific computing and computer graphics an unstructured simplex mesh requires a choice of meshpoints vertex nodes

classification of two term tilting complexes over brauer graph algebras using only the combinatorics of its defining ribbon graph we classify the two term tilting complexes as well as their indecomposable summands of a brauer graph algebra as an application we determine precisely the class of brauer graph algebras which are tilting discrete

a new flux splitting scheme a new flux splitting scheme is proposed the scheme is remarkably simple and yet its accuracy rivals and in some cases surpasses that of roe s solver in the euler and navier stokes solutions carried out in this study the scheme is robust and converges as fast as the roe splitting we propose an appropriately defined cell face advection mach number using values from the two straddling cells via associated characteristic speeds this interface mach number is then used to determine the upwind extrapolation for the convective quantities accordingly the name of the scheme is coined as the advection upstream splitting method ausm we also introduce a new pressure splitting which is shown to behave successfully yielding much smoother results than other existing pressure splittings of particular interest is the supersonic blunt body problem in which the roe scheme gives anomalous solutions the ausm produces correct solutions without difficulty for a wide range of flow conditions as well as grids

a posteriori error estimators for stationary convection diffusion problems a computational comparison abstract we consider the scalar convection diffusion problem in 2 d present standard and stabilized finite element methods and give a survey of different a posteriori error estimators these estimators are either based on the evaluation of local residuals or on the solution of discrete local neumann problems besides that a heuristic indicator based on local averaging techniques for the solution gradient is presented and compared with the estimators we give an overview of another aspect of adaptive mesh refinement namely the decision which elements have to be marked for refinement several marking strategies are described and compared

minimal numerical differentiation formulas we investigate numerical differentiation formulas on irregular centers in two or more variables that are exact for polynomials of a given order and minimize an absolute seminorm of the weight vector error bounds are given in terms of a growth function that carries the information about the geometry of the centers specific forms of weighted ell 1 and weighted least squares minimization are proposed that produce numerical differentiation formulas with particularly good performance in numerical experiments the results are of interest in particular for meshless generalized finite difference methods as they provide a consistency error analysis for such methods

morphisms between indecomposable complexes in the bounded derived category of a gentle algebra in this article we provide a simple combinatorial description of morphisms between indecomposable complexes in the bounded derived category of a gentle algebra

discrete derived categories i homomorphisms autoequivalences and t structures discrete derived categories were studied initially by vossieck j algebra 243 168 176 2001 and later by bobinski et al cent eur j math 2 19 49 2004 in this article we describe the homomorphism hammocks and autoequivalences on these categories we classify silting objects and bounded t structures

adaptive meshless centres and rbf stencils for poisson equation we consider adaptive meshless discretisation of the dirichlet problem for poisson equation based on numerical differentiation stencils obtained with the help of radial basis functions new meshless stencil selection and adaptive refinement algorithms are proposed in 2d numerical experiments show that the accuracy of the solution is comparable with and often better than that achieved by the mesh based adaptive finite element method

preconditioning applied to variable and constant density flows a time derivative preconditioning of the navier stokes equations suitable for both variable and constant density fluids is developed the ideas of low mach number preconditioning and artificial compressibility are combined into a unified approach designed to enhance convergence rates of density based time marching schemes for solving flows of incompressible and variable density fluids at all speeds the preconditioning is coupled with a dual time stepping scheme implemented within an explicit multistage algorithm for solving time accurate flows the resultant time integration scheme is used in conjunction with a finite volume discretization designed for unstructured solution adaptive mesh topologies this method is shown to provide accurate steady state solutions for transonic and low speed flow of variable density fluids the time accurate solution of unsteady incompressible flow is also demonstrated