coercivity hypocoercivity exponential time decay and simulations for discrete fokker planck equations in this article we propose and study several discrete versions of homogeneous and inhomogeneous one dimensional fokker planck equations in particular for these dis cretizations of velocity and space we prove the exponential convergence to the equilibrium of the solutions for time continuous equations as well as for time discrete equations our method uses new types of discrete poincare inequalities for a two direction discretization of the derivative in velocity for the inhomogeneous problem we adapt hypocoercive methods to the discrete cases

classifying blocks with abelian defect groups of rank 3 for the prime 2 abstract in this paper we classify all blocks with defect group c 2 n c 2 c 2 up to morita equivalence together with a recent paper of wu zhang and zhou this completes the classification of morita equivalence classes of 2 blocks with abelian defect groups of rank at most 3 the classification holds for blocks over a suitable discrete valuation ring as well as for those over an algebraically closed field the case considered in this paper is significant because it involves comparison of morita equivalence classes between a group and a normal subgroup of index 2 so requires novel reduction techniques which we hope will be of wider interest we note that this also completes the classification of blocks with abelian defect groups of order dividing 16 up to morita equivalence a consequence is that broue s abelian defect group conjecture holds for all blocks mentioned above

a conservative free energy dissipating and positivity preserving finite difference scheme for multi dimensional nonlocal fokker planck equation abstract in this work we design and analyze a conservative positivity preserving and free energy dissipating finite difference method for the multi dimensional nonlocal fokker planck fp equation based on a non logarithmic landau transformation a central differencing spatial discretization using harmonic mean approximations is developed both forward and backward euler discretizations in time are employed to derive an explicit scheme and a linearized semi implicit scheme respectively three desired properties that are possessed by analytical solutions i mass conservation ii free energy dissipation and iii positivity are proved to be maintained at discrete level remarkably numerical analysis demonstrates that the semi implicit time discretization ensures the property of positivity preserving unconditionally due to the advantages brought by the harmonic mean approximations an estimate on the upper bound of the condition number of the resulting coefficient matrix is further established for the semi implicit scheme extensive numerical tests are performed to validate aforementioned properties numerically

a nonlinear mixed finite eelement method for a degenerate parabolic equation arising in flow in porous media we study a model nonlinear degenerate advection diffusion equation having application in petroleum reservoir and groundwater aquifer simulation the main difficulty is that the true solution is typically lacking in regularity therefore we consider the problem from the point of view of optimal approximation through time integration we develop a mixed variational form that respects the known minimal regularity and then we develop and analyze two versions of a mixed finite element approximation a simpler semidiscrete time continuous version and a fully discrete version our error bounds are optimal in the sense that all but one of the bounding terms reduce to standard approximation error the exceptional term is a nonstandard approximation error term we also consider our new formulation for the nondegenerate problem showing the usual optimal l 2 error bounds moreover superconvergence is obtained under special circumstances

a finite volume scheme for nonlinear degenerate parabolic equations we propose a second order finite volume scheme for nonlinear degenerate parabolic equations which admit an entropy functional for some of these models porous media equation drift diffusion system for semiconductors it has been proved that the transient solution converges to a steady state when time goes to infinity the present scheme preserves steady states and provides a satisfying long time behavior moreover it remains valid and second order accurate in space even in the degenerate case after describing the numerical scheme we present several numerical results which confirm the high order accuracy in various regime degenerate and nondegenerate cases and underline the efficiency to preserve the large time asymptotic

a posteriori error estimation and adaptivity for degenerate parabolic problems two explicit error representation formulas are derived for degenerate parabolic pdes which are based on evaluating a parabolic residual in negative norms the resulting upper bounds are valid for any numerical method and rely on regularity properties of solutions of a dual parabolic problem in nondivergence form with vanishing diffusion coefficient they are applied to a practical space time discretization consisting of c 0 piecewise linear finite elements over highly graded unstructured meshes and backward finite differences with varying time steps two rigorous a posteriori error estimates are derived for this scheme and used in designing an efficient adaptive algorithm which equidistributes space and time discretization errors via refinement coarsening a simulation finally compares the behavior of the rigorous a posteriori error estimators with a heuristic approach and hints at the potentials and reliability of the proposed method

finite element approximation of the transport of reactive solutes in porous media part 1 error estimates for nonequilibrium adsorption processes in this paper we analyze a fully practical piecewise linear finite element approximation involving numerical integration backward euler time discretization and possibly regularization of the following degenerate parabolic system arising in a model of reactive solute transport in porous media find u x t v x t such that begin eqnarray partial t u partial t v delta u f quad mbox in omega times 0 t qquad u 0 quad mbox on partial omega times 0 t partial tv k varphi u v quad mbox in omega times 0 t u cdot 0 g 1 cdot quad v cdot 0 g 2 cdot quad mbox in omega subset real d quad 1 le d le 3 end eqnarray for given data k in real f g 1 g 2 and a monotonically increasing varphi in c 0 real cap c 1 infty 0 cup 0 infty satisfying varphi 0 0 which is only locally holder continuous with exponent p in 0 1 at the origin e g varphi s equiv s p this lack of lipschitz continuity at the origin limits the regularity of the unique solution u v and leads to difficulties in the finite element error analysis

viscous splitting approximation of mixed hyperbolic parabolic convection diffusion equations we first analyse a semi discrete operator splitting method for nonlinear possibly strongly degenerate convection diffusion equations due to strong degeneracy solutions can be discontinuous and are in general not uniquely determined by their data hence weak solutions satisfying an entropy condition are sought we then propose and analyse a fully discrete splitting method which employs a front tracking scheme for the convection step and a finite difference scheme for the diffusion step numerical examples are presented which demonstrate that our method can be used to compute physically correct solutions to mixed hyperbolic parabolic convection diffusion equations

a structure preserving scheme for the kolmogorov fokker planck equation in this paper we introduce a numerical scheme which preserves the behavior of solutions to the kolmogorov equation as time tends to infinity the method presented is based on a self similar change of variables technique to transform the kolmogorov equation into a new form such that the problem of designing structure preserving schemes for the original equation amounts to building a standard scheme for the transformed equation this transformation also has the added benefit of allowing for an exact operator splitting scheme whereas in the original form a standard operator splitting was only second order finally we verify the preservation of long time behavior through numerical simulations

solution of degenerate convection diffusion problems by the method of characteristics a new approximation scheme is proposed to solve nonlinear degenerate convection diffusion problems the method is based on a relaxation scheme developed for degenerate parabolic problems in w jager and j kacur numer math 60 1991 pp 407 427 m2an math model numer anal 29 1995 pp 605 627 and on the method of characteristics j douglas and t i russell siam j numer anal 19 1982 pp 871 885 o pironneau numer math 38 1982 pp 309 332

extensions of rickard equivalences broue conjectures that a block of a finite group g with abelian defect group p is rickard equivalent to its brauer correspondent in the normalizer ng p we find a character theoretic criterion on extensions of rickard equivalences between blocks as an application we prove that the conjecture holds for the principal block of a finite group with abelian defect group and cyclic hyperfocal subgroup

blocks with defect group z2n z2n z2m abstract in this paper we prove that a block algebra with defect group z 2 n z 2 n z 2 m where n 2 and m is arbitrary is morita equivalent to its brauer correspondent

a finite volume scheme for the patlak keller segel chemotaxis model a finite volume method is presented to discretize the patlak keller segel pks modeling chemosensitive movements first we prove existence and uniqueness of a numerical solution to the proposed scheme then we give a priori estimates and establish a threshold on the initial mass for which we show that the numerical approximation converges to the solution to the pks system when the initial mass is lower than this threshold numerical simulations are performed to verify accuracy and the properties of the scheme finally in the last section we investigate blow up of the solution for large mass

a discontinuous galerkin method for nonlinear parabolic equations and gradient flow problems with interaction potentials abstract we consider a class of time dependent second order partial differential equations governed by a decaying entropy the solution usually corresponds to a density distribution hence positivity non negativity is expected this class of problems covers important cases such as fokker planck type equations and aggregation models which have been studied intensively in the past decades in this paper we design a high order discontinuous galerkin method for such problems if the interaction potential is not involved or the interaction is defined by a smooth kernel our semi discrete scheme admits an entropy inequality on the discrete level furthermore by applying the positivity preserving limiter our fully discretized scheme produces non negative solutions for all cases under a time step constraint our method also applies to two dimensional problems on cartesian meshes numerical examples are given to confirm the high order accuracy for smooth test cases and to demonstrate the effectiveness for preserving long time asymptotics

identification of asymptotic decay to self similarity for one dimensional filtration equations the objective of this paper is the derivation and the analysis of a simple explicit numerical scheme for general one dimensional filtration equations it is based on an alternative formulation of the problem using the pseudoinverse of the density s repartition function in particular the numerical approximations can be proven to satisfy a contraction property for a wasserstein metric various numerical results illustrate the ability of this numerical process to capture the time asymptotic decay towards self similar solutions even for fast diffusion equations

high order relaxation schemes for nonlinear degenerate diffusion problems several relaxation approximations to partial differential equations have been recently proposed examples include conservation laws hamilton jacobi equations convection diffusion problems and gas dynamics problems the present paper focuses on diffusive relaxation schemes for the numerical approximation of nonlinear parabolic equations these schemes are based on a suitable semilinear hyperbolic system with relaxation terms high order methods are obtained by coupling eno and weighted essentially nonoscillatory weno schemes for space discretization with implicit explicit imex schemes for time integration error estimates and a convergence analysis are developed for semidiscrete schemes with a numerical analysis for fully discrete relaxed schemes various numerical results in one and two dimensions illustrate the high accuracy and good properties of the proposed numerical schemes also in the degenerate case these schemes can be easily implemented on parallel computers and applied to more general systems of nonlinear parabolic equations in two and three dimensional cases

a combined finite volume nonconforming mixed hybrid finite element scheme for degenerate parabolic problems we propose and analyze a numerical scheme for nonlinear degenerate parabolic convection diffusion reaction equations in two or three space dimensions we discretize the diffusion term which generally involves an inhomogeneous and anisotropic diffusion tensor over an unstructured simplicial mesh of the space domain by means of the piecewise linear nonconforming crouzeix raviart finite element method or using the stiffness matrix of the hybridization of the lowest order raviart thomas mixed finite element method the other terms are discretized by means of a cell centered finite volume scheme on a dual mesh where the dual volumes are constructed around the sides of the original mesh checking the local peclet number we set up the exact necessary amount of upstream weighting to avoid spurious oscillations in the convection dominated case this technique also ensures the validity of the discrete maximum principle under some conditions on the mesh and the diffusion tensor we prove the convergence of the scheme only supposing the shape regularity condition for the original mesh we use a priori estimates and the kolmogorov relative compactness theorem for this purpose the proposed scheme is robust only 5 point 7 point in space dimension three locally conservative efficient and stable which is confirmed by numerical experiments

affirmative answer to a question of feit as he points out if p is abelian the affirmative answer follows from dade s classification 4 our purpose here is to show that the general case reduces to the abelian one which answers affirmatively feit s question more generally in 4 dade determines the cap group associated with a finite abelian p group showing in particular that this group is finitely generated our main result here cf theorem 2 2 below implies that finite generation is still true in the non abelian case cf corollary 2 4 below this result and the main steps of a proof were already announced in 17 section81 and here we are just discharging ourselves of an old debt actually as we said in 7 section81 our result would follow from a complete classification of endo permutation modules

error analysis for characteristics based methods for degenerate parabolic problems we consider characteristics based finite element methods for solving nonlinear degenerate advection diffusion equations these equations have applications in the simulation of petroleum reservoirs and groundwater aquifers and in the modeling of free boundary problems standard finite element galerkin methods have been studied for these equations in this paper we analyze the characteristics based finite element methods for them the main difficulty in the analysis is that the equations are degenerate and the solution lacks regularity here we develop a technique that respects the degeneracy and the known minimal regularity this technique is based on the green operator for standard elliptic equations and is developed directly for the degenerate advection diffusion equations we concentrate our analysis on the modified method of characteristics mmoc and one of its variants the modified method of characteristics with adjusted advection mmocaa which conserves mass we derive error estimates in various norms the extension to other variants is discussed the present technique is also applied to nondegenerate problems error estimates previously obtained for the mmoc are derived under much weaker regularity assumptions on the solution and the error estimates for the mmocaa appear new even in the nondegenerate case finally numerical results are presented to show the sharpness of the error estimates derived

a finite volume scheme for convection diffusion equations with nonlinear diffusion derived from the scharfetter gummel scheme we propose a finite volume scheme for convection diffusion equations with nonlinear diffusion such equations arise in numerous physical contexts we will particularly focus on the drift diffusion system for semiconductors and the porous media equation in these two cases it is shown that the transient solution converges to a steady state solution as t tends to infinity the introduced scheme is an extension of the scharfetter gummel scheme for nonlinear diffusion it remains valid in the degenerate case and preserves steady states we prove the convergence of the scheme in the nondegenerate case finally we present some numerical simulations applied to the two physical models introduced and we underline the efficiency of the scheme to preserve long time behavior of the solutions

a second order positivity preserving numerical method for gamma equation in this work we consider cauchy problem for the so called gamma equation which can be derived by transforming the fully nonlinear black scholes equation for option price into a quasi linear parabolic equation for the second derivative greek c v s s of the option price v we develop an efficient positivity preserving explicit numerical method for solving the model problem concerning different volatility terms we prove that the obtained semi discretization is positive and the corresponding full approximation also preserves this property if the time step is restricted the stability of the difference scheme in l 1 norm is shown and the existence of interface curves is investigated numerically results of numerical simulations are given and discussed

solution of porous medium type systems by linear approximation schemes the convergence of semi discrete and discrete linear approximation schemes is analysed for nonlinear degenerate parabolic systems of porous medium type the enthalpy formulation and variational technique are used the semi discretization used reduces the original parabolic p d e to linear elliptic p d e the algebraic correction arising from nonlinearities is treated by newton like iterations in finite steps some numerical experiments are discussed and compared with the analytical solutions

2 blocks with abelian defect groups we give a classification up to morita equivalence of 2 blocks of quasi simple groups with abelian defect groups as a consequence we show that donovan s conjecture holds for elementary abelian 2 groups and that the entries of the cartan matrices are bounded in terms of the defect for arbitrary abelian 2 groups we also show that a block with defect groups of the form c2m c2m for m 2 has one of two morita equivalence types and hence is morita equivalent to the brauer correspondent block of the normaliser of a defect group this completes the analysis of the morita equivalence types of 2 blocks with abelian defect groups of rank 2 from which we conclude that donovan s conjecture holds for such 2 groups a further application is the completion of the determination of the number of irreducible characters in a block with abelian defect groups of order 16 the proof uses the classification of finite simple groups

fermi dirac fokker planck equation well posedness long time asymptotics abstract a fokker planck type equation for interacting particles with exclusion principle is analyzed the nonlinear drift gives rise to mathematical difficulties in controlling moments of the distribution function assuming enough initial moments are finite we can show the global existence of weak solutions for this problem the natural associated entropy of the equation is the main tool to derive uniform in time a priori estimates for the kinetic energy and entropy as a consequence long time asymptotics in l 1 are characterized by the fermi dirac equilibrium with the same initial mass this result is achieved without rate for any constructed global solution and with exponential rate due to entropy entropy dissipation arguments for initial data controlled by fermi dirac distributions finally initial data below radial solutions with suitable decay at infinity lead to solutions for which the relative entropy towards the fermi dirac equilibrium is shown to converge to zero without decay rate

on the optimal exercise boundaries of swing put options we use probabilistic methods to characterise time dependent optimal stopping boundaries in a problem of multiple optimal stopping on a finite time horizon motivated by financial applications we consider a payoff of immediate stopping of put type and the underlying dynamics follows a geometric brownian motion the optimal stopping region relative to each optimal stopping time is described in terms of two boundaries which are continuous monotonic functions of time and uniquely solve a system of coupled integral equations of volterra type finally we provide a formula for the value function of the problem

maximum principle satisfying third order discontinuous galerkin schemes for fokker planck equations we design and analyze up to third order accurate discontinuous galerkin dg methods satisfying a strict maximum principle for fokker planck equations a procedure is established to identify an effective test set in each computational cell to ensure the desired bounds of numerical averages during time evolution this is achievable by taking advantage of the two parameters in the numerical flux and a novel decomposition of weighted cell averages based on this result a scaling limiter for the dg method with first order euler forward time discretization is proposed to solve the one dimensional fokker planck equations strong stability preserving high order time discretizations will keep the maximum principle it is straightforward to extend the method to two and higher dimensions on rectangular meshes we also show that a modified limiter can preserve the strict maximum principle for dg schemes solving fokker planck equations as a consequence the present schemes preserve steady states numerical tests

convergence of a nonlinear entropy diminishing control volume finite element scheme for solving anisotropic degenerate parabolic equations in this paper we propose and analyze a control volume finite elements cvfe scheme for solving possibly degenerated parabolic equations this scheme does not require the introduction of the so called kirchhoff transform in its definition we prove that the discrete solution obtained emph via the scheme remains in the physical range and that the natural entropy of the problem decreases with time the convergence of the method is proved as the discretization steps tend to 0 finally numerical examples illustrate the efficiency of the method

finite volume methods for convection diffusion problems derivation stability and error analysis in both discrete h 1 and l 2 norms for cell centered finite volume approximations of convection diffusion problems are presented various upwind strategies are investigated the theoretical results are illustrated by numerical examples

entropy decay of discretized fokker planck equations i temporal semidiscretization abstract in this paper we study the large time behavior of a fully implicit semidiscretization in time of parabolic fokker planck type equations using logarithmic sobolev inequalities exponential decay of the relative entropy w r t the steady state is proved which yields convergence of the discrete scheme towards the unique steady state the exponential decay rate recovers as at j 0 the decay rate of the original fokker planck type equations

explicit diffusive kinetic schemes for nonlinear degenerate parabolic systems we design numerical schemes for nonlinear degenerate parabolic systems with possibly dominant convection these schemes are based on discrete bgk models where both characteristic velocities and the source term depend singularly on the relaxation parameter general stability conditions are derived and convergence is proved to the entropy solutions for scalar equations

the role of a strong confining potential in a nonlinear fokker planck equation abstract we show that solutions of nonlinear nonlocal fokker planck equations in a bounded domain with no flux boundary conditions can be approximated by cauchy problems with increasingly strong confining potentials defined in the whole space two different approaches are analyzed making crucial use of uniform estimates for l 2 energy functionals and free energy or entropy functionals respectively in both cases we prove that the weak formulation of the problem in a bounded domain can be obtained as the weak formulation of a limit problem in the whole space involving a suitably chosen sequence of large confining potentials the free energy approach extends to the case degenerate diffusion

a finite volume scheme for boundary driven convection diffusion equations with relative entropy structure we propose a finite volume scheme for a class of nonlinear parabolic equations endowed with non homogeneous dirichlet boundary conditions and which admit relative entropy functionals for this kind of models including the porous media equations fokker planck equations for plasma physics or dumbbell models for polymer flows it has been proved that the transient solution converges to a steady state when time goes to infinity the present scheme is built from the resolution of the stationary equation in order to preserve steady states and natural lyapunov functionals which provide a satisfying long time behavior after describing the numerical scheme we present several numerical results which confirm the accuracy and underline the efficiency to preserve the large time asymptotic

a numerical approach to degenerate parabolic equations in this work we propose a numerical approach to solve some kind of degenerate parabolic equations the underlying idea is based on the maximum principle more precisely we locally perturb the initial and boundary data instead of the nonlinear diffusion coefficients so that the resulting problem is not degenerate the efficiency of this method is shown analytically as well as numerically the numerical experiments show that this new approach is comparable with the existing ones

on the alperin mckay conjecture for simple groups of type a abstract in this paper characters of the normalizer of d split levi subgroups in sl n q and su n q are parametrized with a particular focus on the clifford theory between the levi subgroup and its normalizer these results are applied to verify the alperin mckay conjecture for primes l with l 6 q 2 1 and the alperin weight conjecture for l blocks of those quasi simple groups with abelian defect the inductive alperin mckay condition and inductive alperin weight condition by the second author are verified for certain blocks of sl n q and su n q

convergence of a finite volume scheme for a system of interacting species with cross diffusion in this work we present the convergence of a positivity preserving semi discrete finite volume scheme for a coupled system of two non local partial differential equations with cross diffusion the key to proving the convergence result is to establish positivity in order to obtain a discrete energy estimate to obtain compactness we numerically observe the convergence to reference solutions with a first order accuracy in space moreover we recover segregated stationary states in spite of the regularising effect of the self diffusion however if the self diffusion or the cross diffusion is strong enough mixing occurs while both densities remain continuous

convergence of a finite volume scheme for nonlinear degenerate parabolic equations one approximates the entropy weak solution u of a nonlinear parabolic degenerate equation u t rm div mathbf q f u delta phi u 0 by a piecewise constant function u mathcal d using a discretization mathcal d in space and time and a finite volume scheme the convergence of u mathcal d to u is shown as the size of the space and time steps tend to zero in a first step estimates on u mathcal d are used to prove the convergence up to a subsequence of u mathcal d to a measure valued entropy solution called here an entropy process solution a result of uniqueness of the entropy process solution is proved yielding the strong convergence of u mathcal d to it u some on a model equation are shown

approximation of degenerate parabolic problems using numerical integration a class of multidimensional degenerate parabolic equations is considered the two phase stefan problem and the porous medium equation are analyzed as models of singular parabolic equations nonstat

error estimates for a class of degenerate parabolic equations the degenerate parabolic equation u t delta beta u is considered in the cylinder omega times 0 t where omega subset r n is bounded beta 0 beta 0 0 and beta geq0 e g beta u u u m 1 m 1 a fully discrete scheme is given using c0 piecewise linear finite elements in space and a backward difference scheme in time l2 error estimates for the approximation of u and of beta u are proven for that purpose we employ the fact that the free boundary has a finite speed of propagation

2 blocks whose defect group is homocyclic and whose inertial quotient contains a singer cycle abstract we consider a block b of a finite group with defect group d c 2 m n and inertial quotient e containing a singer cycle an element of order 2 n 1 this implies e e f where e c 2 n 1 f c n and e acts transitively on the elements in d of order 2 we classify the basic morita equivalence classes of b over a complete discrete valuation ring o when m 1 b is basic morita equivalent to the principal block of one of s l 2 2 n f d e or j 1 where j 1 occurs only when n 3 when m 1 b is basic morita equivalent to d e