a fast spectral element solver combining static condensation and multigrid techniques we propose a spectral element multigrid method for the two dimensional helmholtz equation discretized on regular grids combining p multigrid with static condensation the method achieves nearly linear complexity with an order independent convergence rate for solving the condensed equations for smoothing we consider two groups of edge based relaxation schemes the best of which attains a multigrid convergence rate of 0 014 to 0 028 numerical experiments have been carried out that demonstrate the robustness of the approach for orders up to 32 and a total of 109 degrees of freedom in comparison with a fast finite difference solver the latter is clearly outperformed already for errors of one percent or lower

long time regularity of solutions of the hele shaw problem abstract in this paper we study the long time behavior of solutions of the one phase hele shaw problem without surface tension we show that after a finite time solutions of the hele shaw problem become starshaped and lipschitz continuous in space based on this observation we then prove that the free boundary become smooth in space and time with nondegenerate free boundary speed

uniqueness and existence results on the hele shaw and the stefan problems in this paper we introduce a notion of viscosity solutions for the one phase hele shaw and stefan problems when there is no surface tension we prove the uniqueness and existence of the solutions for both problems and the uniform convergence of solutions of porous medium equation to those of the hele shaw problem

a p multigrid discontinuous galerkin method for the euler equations on unstructured grids a p multigrid p polynomial degree discontinuous galerkin method is presented for the solution of the compressible euler equations on unstructured grids the method operates on a sequence of solution approximations of different polynomial orders a distinct feature of this p multigrid method is to use different time integration schemes on different approximation levels resulting in an accurate fast and low memory method that can be used to accelerate the convergence of the euler equations to a steady state for discontinuous galerkin methods the developed method is used to compute the compressible flows for a variety of test problems on unstructured grids the numerical results obtained strongly indicate the order independent property of this p multigrid method an overall speed up factor more than one order of magnitude for both second and third order solutions of all test cases in comparison with the explicit method is demonstrated

a space time discontinuous galerkin method for the incompressible navier stokes equations we introduce a space time discontinuous galerkin dg finite element method for the incompressible navier stokes equations our formulation can be made arbitrarily high order accurate in both space and time and can be directly applied to deforming domains different stabilizing approaches are discussed which ensure stability of the method a numerical study is performed to compare the effect of the stabilizing approaches to show the method s robustness on deforming domains and to investigate the behavior of the convergence rates of the solution recently we introduced a space time hybridizable dg hdg method for incompressible flows s rhebergen b cockburn a space time hybridizable discontinuous galerkin method for incompressible flows on deforming domains j comput phys 231 2012 4185 4204 we will compare numerical results of the space time dg and space time hdg methods this constitutes the first comparison between dg and hdg methods

coupling p multigrid to geometric multigrid for discontinuous galerkin formulations of the convection diffusion equation an improved p multigrid algorithm for discontinuous galerkin dg discretizations of convection diffusion problems is presented the general p multigrid algorithm for dg discretizations involves a restriction from the p 1 to p 0 discontinuous polynomial solution spaces this restriction is problematic and has limited the efficiency of the p multigrid method for purely diffusive problems helenbrook and atkins have demonstrated rapid convergence using a method that restricts from a discontinuous to continuous polynomial solution space at p 1 it is shown that this method is not directly applicable to the convection diffusion cd equation because it results in a central difference discretization for the convective term to remedy this ideas from the streamwise upwind petrov galerkin supg formulation are used to devise a transition from the discontinuous to continuous space at p 1 that yields an upwind discretization the results show that the new method converges rapidly for all peclet numbers

a variational inequality approach to hele shaw flow with a moving boundary this paper is concerned with the study of a mathematical model of the injection of fluid into a finite hele shaw cell the mathematical problem is one of solving laplace s equation in an unknown region whose boundary changes with time by a transformation of the dependent variable an elliptic variational inequality formulation of the moving boundary problem is obtained the variational inequality is shown to have a unique solution up to the time at which the cell is filled regularity results for the solution of the inequality are obtained by studying a penalty approximation of the inequality

robust multigrid for high order discontinuous galerkin methods we present a polynomial multigrid method for nodal interior penalty and local discontinuous galerkin formulations of the poisson equation on cartesian grids for smoothing we propose two classes of overlapping schwarz methods the first class comprises element centered and the second face centered methods within both classes we identify methods that achieve superior convergence rates prove robust with respect to the mesh spacing and the polynomial order at least up to p 32 consequent structure exploitation yields a computational complexity of o pn where n is the number of unknowns further we demonstrate the suitability of the face centered method for element aspect ratios up to 32 p multigrid for discontinuous formulations of poisson problems on cartesian grids robust with respect to the mesh size and polynomial order at least up to p 32 convergence rates between 0 02 and 0 003 improving with increasing p computational complexity of pn for n unknowns linear complexity in runtime excellent performance up to aspect ratios of 1 16

space time discontinuous galerkin finite element method with dynamic grid motion for inviscid compressible flows i general formulation a new space time discontinuous galerkin finite element method for the solution of the euler equations of gas dynamics in time dependent flow domains is presented the discontinuous galerkin discretization results in an efficient elementwise conservative upwind finite element method which is particularly well suited for local mesh refinement the upwind scheme uses a formulation of the hllc flux applicable to moving meshes and several formulations for the stabilization operator to ensure that monotone solutions around discontinuities are investigated the non linear equations of the space time discretization are solved using a multigrid accelerated pseudo time integration technique with an optimized runge kutta method the linear stability of the pseudo time integration method is investigated for the linear advection equation the numerical scheme is demonstrated with simulations of the flow field in a shock tube a channel with a bump and an oscillating naca 0012 airfoil these simulations show that using the data at the superconvergence points the accuracy of the numerical discretization is o h5 2 in space for smooth subsonic flows both on structured and on locally refined meshes and that the space time adaptation can significantly improve the accuracy and efficiency of the numerical method

h multigrid for space time discontinuous galerkin discretizations of the compressible navier stokes equations being implicit in time the space time discontinuous galerkin discretization of the compressible navier stokes equations requires the solution of a non linear system of algebraic equations at each time step the overall performance therefore highly depends on the efficiency of the solver in this article we solve the system of algebraic equations with a h multigrid method using explicit runge kutta relaxation two level fourier analysis of this method for the scalar advection diffusion equation shows convergence factors between 0 5 and 0 75 this motivates its application to the 3d compressible navier stokes equations where numerical experiments show that the computational effort is significantly reduced up to a factor 10 w r t single grid iterations

regularity of the free boundary in parabolic phase transition problems in this paper we start the study of the regularity properties of the free boundary for parabolic two phase free boundary problems may be the best known example of a parabolic two phase free boundary problem is the stefan problem a simplified model describing the melting or solidification of a material with a solid liquid interphase the concept of solution can be stated in several ways classical solution weak so lution on divergence form or viscosity solution and as usual one would like to prove that the weak solutions that may be constructed are in fact as smooth and classical as possible locally a classical solution of the stefan problem may be described as following on the unit cylinder q1 b1 1 1 we have two complementary domains and qi separated by a smooth surface s oi2 nq1 in fl and qi we have two smooth solutions ul and u2 of the heat equations

hp multigrid as smoother algorithm for higher order discontinuous galerkin discretizations of advection dominated flows part i multilevel analysis the hp multigrid as smoother algorithm hp mgs for the solution of higher order accurate space time discontinuous galerkin discretizations of advection dominated flows is presented this algorithm combines p multigrid with h multigrid at all p levels where the h multigrid acts as smoother in the p multigrid the performance of the hp mgs algorithm is further improved using semi coarsening in combination with a new semi implicit runge kutta method as smoother a detailed multilevel analysis of the hp mgs algorithm is presented to obtain more insight into the theoretical performance of the algorithm as model problem a fourth order accurate space time discontinuous galerkin discretization of the advection diffusion equation is considered the multilevel analysis shows that the hp mgs algorithm has excellent convergence rates both for steady state and time dependent problems and low and high cell reynolds numbers including highly stretched meshes

finite flatness of torsion subschemes of hilbert blumenthal abelian varieties let e be a totally real number field of degree d over q we give a method for constructing a set of hilbert modular cuspforms f1 fd with the following property let k be the fraction field of a complete dvr a and let x k be a hilbert blumenthal abelian variety with multiplicative reduction and real multiplication by the ring of integers of e suppose n is an integer such that n divides the minimal valuation of fi x for all i then x n k extends to a finite flat group scheme over a where n is a divisor of n with n n bounded by a constant depending only on f1 fd when e q the theorem reduces to a well known property of f1 in the cases e q 2 and e q 5 we produce the desired pairs of hilbert modular forms explicitly and show how they can be used to compute the group of neron components of a hilbert blumenthal abelian variety with real multiplication by e

application of p multigrid to discontinuous galerkin formulations of the poisson equation we investigate p multigrid as a solution method for several different discontinuous galerkin dg formulations of the poisson equation different combinations of relaxation schemes and basis sets have been combined with the dg formulations to find the best performing combination the damping factors of the schemes have been determined using fourier analysis for both one and two dimensional problems one important finding is that when using dg formulations the standard approach of forming the coarse p matrices separately for each level of multigrid is often unstable to ensure stability the coarse p matrices must be constructed from the fine grid matrices using algebraic multigrid techniques of the relaxation schemes we find that the combination of jacobi relaxation with the spectral element basis is fairly effective the results using this combination are p sensitive in both one and two dimensions but reasonable convergence rates can still be achieved for moderate values of p and isotropic meshes a competitive alternative is a block gauss seidel relaxation this actually out performs a more expensive line relaxation when the mesh is isotropic when the mesh becomes highly anisotropic the implicit line method and the gauss seidel implicit line method are the only effective schemes adding the gauss seidel terms to the implicit line method gives a significant improvement over the line relaxation method

hp multigrid as smoother algorithm for higher order discontinuous galerkin discretizations of advection dominated flows part ii optimization of the runge kutta smoother using a detailed multilevel analysis of the complete hp multigrid as smoother algorithm accurate predictions are obtained of the spectral radius and operator norms of the multigrid error transformation operator this multilevel analysis is used to optimize the coefficients in the semi implicit runge kutta smoother such that the spectral radius of the multigrid error transformation operator is minimal under properly chosen constraints the runge kutta coefficients for a wide range of cell reynolds numbers and a detailed analysis of the performance of the hp mgs algorithm are presented in addition the computational complexity of the hp mgs algorithm is investigated the hp mgs algorithm is tested on a fourth order accurate space time discontinuous galerkin finite element discretization of the advection diffusion equation for a number of model problems which include thin boundary layers and highly stretched meshes and a non constant advection velocity for all test cases excellent multigrid convergence is obtained

regularity of the free boundary for the one phase hele shaw problem abstract in this paper we prove that in a local neighborhood lipschitz continuous free boundary of a solution of the one phase hele shaw problem is indeed smooth if the solution is lipschitz continuous and non degenerate in the neighborhood

a hele shaw problem for tumor growth abstract we consider weak solutions to a problem modeling tumor growth under certain conditions on the initial data solutions can be obtained by passing to the stiff incompressible limit in a porous medium type problem with a lotka volterra source term describing the evolution of the number density of cancerous cells we prove that such limit solutions solve a free boundary problem of hele shaw type we also obtain regularity properties both for the solution and for its free boundary the main new difficulty arises from the competition between the growth due to the source which keeps the initial singularities and the free boundary which invades the domain with a regularizing effect new islands can be generated at singular times