generalized surface quasi geostrophic equations with singular velocities this paper establishes several existence and uniqueness results for two families of active scalar equations with velocity fields determined by the scalars through very singular integrals the first family is a generalized surface quasigeostrophic sqg equation with the velocity field u related to the scalar by where and is the zygmund operator the borderline case 1 corresponds to the sqg equation and the situation is more singular for 1 we obtain the local existence and uniqueness of classical solutions the global existence of weak solutions and the local existence of patch type solutions the second family is a dissipative active scalar equation with which is at least logarithmically more singular than the velocity in the first family we prove that this family with any fractional dissipation possesses a unique local smooth solution for any given smooth data this result for the second family constitutes a first step towards resolving the global regularity issue recently proposed by k ohkitani 2012 wiley periodicals inc

on global regularity of 2d generalized magnetohydrodynamic equations abstract in this article we study the global regularity of 2d generalized magnetohydrodynamic equations 2d gmhd in which the dissipation terms are u and b we show that smooth solutions are global in the following three cases 1 2 1 0 1 2 2 2 2 0 we also show that in the inviscid case 0 if 1 then smooth solutions are global as long as the direction of the magnetic field remains smooth enough

the well posedness of the surface quasi geostrophic equations in the besov morrey spaces abstract in this paper we develop the boundedness of riesz transform in morrey spaces which generalize the corresponding result in the usual l p spaces as an application of this boundedness we establish the well posedness and blow up criterion for the surface quasi geostrophic equations in the framework of besov morrey spaces

lifespan of solutions to semilinear damping wave equations in de sitter spacetime abstract this paper studies blowup phenomena for two kind of initial value problem of semilinear damping wave equations in de sitter spacetime the first problem comes from the latter problem of conjecture of strauss in de sitter spacetime and the second problem arises from the latter problem of conjecture of glassey in de sitter spacetime we give blowup and lifespan of solutions on those two problems comparing with non damping case established in yan 2018 one can see that the damping term with a positive bounded time dependent coefficient does not effect blowup of solutions meanwhile we establish a new ode blowup result concerning damping term

unified hybridization of discontinuous galerkin mixed and continuous galerkin methods for second order elliptic problems we introduce a unifying framework for hybridization of finite element methods for second order elliptic problems the methods fitting in the framework are a general class of mixed dual finite element methods including hybridized mixed continuous galerkin nonconforming and a new wide class of hybridizable discontinuous galerkin methods the distinctive feature of the methods in this framework is that the only globally coupled degrees of freedom are those of an approximation of the solution defined only on the boundaries of the elements since the associated matrix is sparse symmetric and positive definite these methods can be efficiently implemented moreover the framework allows in a single implementation the use of different methods in different elements or subdomains of the computational domain which are then automatically coupled finally the framework brings about a new point of view thanks to which it is possible to see how to devise novel methods displaying very localized and simple mortaring techniques as well as methods permitting an even further reduction of the number of globally coupled degrees of freedom

dissipative quasi geostrophic equation for large initial data in the critical sobolev space the critical and super critical dissipative quasi geostrophic equations are investigated in mathbb r 2 we prove local existence of a unique regular solution for arbitrary initial data in h2 2 which corresponds to the scaling invariant space of the equation we also consider the behavior of the solution near t 0 in the sobolev space

optimal discrete transmission conditions for a nonoverlapping domain decomposition method for the helmholtz equation this paper is dedicated to recent developments of a two lagrange multipliers domain decomposition method for the helmholtz equation c farhat et al comput methods appl mech engrg 184 2000 pp 213 240 m j gander f magoules and f nataf siam j sci comput 24 2002 pp 38 60 involving an additional augmented operator along the interface between the subdomains most methods for optimizing the augmented interface operator are based on the discretization of approximations of the continuous transparent operator b despres proceedings of the second international conference on mathematical and numerical aspects of wave propagation r kleinman et al eds siam philadelphia 1993 pp 197 206 j d benamou and b despres j comput phys 136 1997 pp 68 82 p chevalier and f nataf domain decomposition methods 10 ams providence ri 1998 pp 400 407 m j gander proceedings of the 12th international conference on domain decomposition methods chiba japan ddm org 2000 pp 247 254 m j gander f magoules and f nataf siam j sci comput 24 2002 pp 38 60 at the discrete level the optimal operator can be proved to be equal to the schur complement of the outer domain this schur complement can be directly approximated using purely algebraic techniques like sparse approximate inverse methods or incomplete factorization the main advantage of such an algebraic approach is that it is much easier to implement in existing code without any information on the geometry of the interface and the finite element formulation used convergence results and parallel efficiency of several algebraic optimization techniques of an interface operator for acoustic analysis applications will be presented

combined effects of singular and sublinear nonlinearities in some elliptic problems abstract we consider a parametric singular dirichlet equation with the singular term u appearing in the left hand side we establish the existence and nonexistence of positive solutions as the parameter 0 and the exponent 0 of the singularity vary in particular we show that for all 0 and all 1 the problem has no positive solution our approach combines truncation arguments with the method of upper and lower solutions

global well posedness for the critical 2d dissipative quasi geostrophic equation we give an elementary proof of the global well posedness for the critical 2d dissipative quasi geostrophic equation the argument is based on a non local maximum principle involving appropriate moduli of continuity

a high order non conforming discontinuous galerkin method for time domain electromagnetics in this paper we discuss the formulation stability and validation of a high order non dissipative discontinuous galerkin dg method for solving maxwell s equations on non conforming simplex meshes the proposed method combines a centered approximation for the numerical fluxes at inter element boundaries with either a second order or a fourth order leap frog time integration scheme moreover the interpolation degree is defined at the element level and the mesh is refined locally in a non conforming way resulting in arbitrary level hanging nodes the method is proved to be stable and conserves a discrete counterpart of the electromagnetic energy for metallic cavities numerical experiments with high order elements show the potential of the method

a new bernstein s inequality and the 2d dissipative quasi geostrophic equation we show a new bernstein s inequality which generalizes the results of cannone planchon danchin and lemarie rieusset as an application of this inequality we prove the global well posedness of the 2d quasi geostrophic equation with the critical and super critical dissipation for the small initial data in the critical besov space and local well posedness for the large initial data

a hybridizable discontinuous galerkin method combined to a schwarz algorithm for the solution of 3d time harmonic maxwell s equation a schwarz type domain decomposition method is presented for the solution of the system of 3d time harmonic maxwell s equations we introduce a hybridizable discontinuous galerkin hdg scheme for the discretization of the problem based on a tetrahedrization of the computational domain the discrete system of the hdg method on each subdomain is solved by an optimized sparse direct lu factorization solver the solution of the interface system in the domain decomposition framework is accelerated by a krylov subspace method the formulation and the implementation of the resulting dd hdg domain decomposed hybridizable discontinuous galerkin method are detailed numerical results show that the resulting dd hdg solution strategy has an optimal convergence rate and can save both cpu time and memory cost compared to a classical upwind flux based dd dg domain decomposed discontinuous galerkin approach

nodal high order methods on unstructured grids we present a convergent high order accurate scheme for the solution of linear conservation laws in geometrically complex domains as our main example we include a detailed development and analysis of a scheme for the time domain solution of maxwell s equations in a three dimensional domain the fully unstructured spatial discretization is made possible by the use of a high order nodal basis employing multivariate lagrange polynomials defined on the triangles and tetrahedra while the equations themselves are satisfied in a discontinuous galerkin form with the boundary conditions being enforced weakly through a penalty term accuracy stability and convergence of the semidiscrete approximation to maxwell s equations is established rigorously and bounds on the growth of the global divergence error are provided concerns related to efficient implementations are discussed in detail this sets the stage for the presentation of examples verifying the theoretical results and illustrating the versatility flexibility and robustness when solving two and three dimensional benchmark problems in computational electromagnetics pure scattering as well as penetration is discussed and high parallel performance of the scheme is demonstrated

global well posedness in the super critical dissipative quasi geostrophic equations we consider the quasi geostrophic equation with the dissipation term in the case constantin cordoba wu 6 proved the global existence of strong solution in h 1 and h 2 under the assumption of small l norm of initial data in this paper we prove the global existence in the scale invariant besov space b 2 2 2 1 for initial data small in the b 2 2 2 1 norm we also prove a global stability result in b 1 2 1

locally implicit discontinuous galerkin method for time domain electromagnetics in the recent years there has been an increasing interest in discontinuous galerkin time domain dgtd methods for the solution of the unsteady maxwell equations modeling electromagnetic wave propagation one of the main features of dgtd methods is their ability to deal with unstructured meshes which are particularly well suited to the discretization of the geometrical details and heterogeneous media that characterize realistic propagation problems such dgtd methods most often rely on explicit time integration schemes and lead to block diagonal mass matrices however explicit dgtd methods are also constrained by a stability condition that can be very restrictive on highly refined meshes and when the local approximation relies on high order polynomial interpolation an implicit time integration scheme is a natural way to obtain a time domain method which is unconditionally stable but at the expense of the inversion of a global linear system at each time step a more viable approach consists of applying an implicit time integration scheme locally in the refined regions of the mesh while preserving an explicit time scheme in the complementary part resulting in an hybrid explicit implicit or locally implicit time integration strategy in this paper we report on our recent efforts towards the development of such a hybrid explicit implicit dgtd method for solving the time domain maxwell equations on unstructured simplicial meshes numerical experiments for 3d propagation problems in homogeneous and heterogeneous media illustrate the possibilities of the method for simulations involving locally refined meshes

a reduced order discontinuous galerkin method based on pod for electromagnetic simulation this paper is concerned with the design of a reduced order discontinuous galerkin dg method based on the proper orthogonal decomposition pod method for electromagnetic simulation a centered flux approximation for surface integral and a second order leap frog scheme for advancing in time are applied in the classical dg method the pod basis is created by the eigensystem of the correlation matrix which is generated by the snapshot matrix whose columns are the snapshot vectors extracted from the high fidelity dg simulation the pod discontinuous galerkin time domain formulation with lower dimension and sufficiently high accuracy is established by applying a galerkin projection for the semidiscrete dg scheme the overall goal is to reduce the computational cost while maintaining an acceptable level of accuracy numerical experiments for electromagnetic problems illustrate the performance of the proposed reduction method

combined effects of singular and superlinear nonlinearities in some singular boundary value problems abstract this paper deals with a class of singular semilinear elliptic dirichlet boundary value problems where the combined effects of a superlinear and a singular term allow us to establish some existence and multiplicity results

parallel multilevel series k way partitioning scheme for irregular graphs in this paper we present a parallel formulation of a multilevel k way graph partitioning algorithm a key feature of this parallel formulation is that it is able to achieve a high degree of concurrency while maintaining the high quality of the partitions produced by the serial multilevel k way partitioning algorithm in particular the time taken by our parallel graph partitioning algorithm is only slightly longer than the time taken for re arrangement of the graph among processors according to the new partition experiments with a variety of finite element graphs show that our parallel formulation produces high quality partitionings in a short amount of time for example a 128 way partitioning of graphs with one million vertices can be computed in a little over two seconds on a 128 processor cray t3d furthermore the quality of the partitions produced is comparable edge cuts within 5 to those produced by the serial multilevel k way algorithm thus our parallel algorithm makes it feasible to perform frequent repartitioning of graphs in dynamic computations without compromising the partitioning quality

mixed discontinuous galerkin approximation of the maxwell operator we introduce and analyze a discontinuous galerkin discretization of the maxwell operator in mixed form here all the unknowns of the underlying system of partial differential equations are approximated by discontinuous finite element spaces of the same order for piecewise constant coefficients the method is shown to be stable and optimally convergent with respect to the mesh size numerical experiments highlighting the performance of the proposed method for problems with both smooth and singular analytical solutions are presented

hybridizable discontinuous galerkin methods for the time harmonic maxwell s equations we present two hybridizable discontinuous galerkin hdg methods for the numerical solution of the time harmonic maxwell s equations the first hdg method explicitly enforces the divergence free condition and thus necessitates the introduction of a lagrange multiplier it produces a linear system for the degrees of freedom of the approximate traces of both the tangential component of the vector field and the lagrange multiplier the second hdg method does not explicitly enforce the divergence free condition and thus results in a linear system for the degrees of freedom of the approximate trace of the tangential component of the vector field only for both hdg methods the approximate vector field converges with the optimal order of k 1 in the l 2 norm when polynomials of degree k are used to represent all the approximate variables we propose elementwise postprocessing to obtain a new h c u r l conforming approximate vector field which converges with order k 1 in the h c u r l norm we present extensive numerical examples to demonstrate and compare the performance of the hdg methods

optimized schwarz methods for maxwell s equations over the last two decades classical schwarz methods have been extended to systems of hyperbolic partial differential equations using characteristic transmission conditions and it has been observed that the classical schwarz method can be convergent even without overlap in certain cases this is in strong contrast to the behavior of classical schwarz methods applied to elliptic problems for which overlap is essential for convergence more recently optimized schwarz methods have been developed for elliptic partial differential equations these methods use more effective transmission conditions between subdomains than the classical dirichlet conditions and optimized schwarz methods can be used both with and without overlap for elliptic problems we show here why the classical schwarz method applied to both the time harmonic and time discretized maxwell s equations converges without overlap the method has the same convergence factor as a simple optimized schwarz method for a scalar elliptic equation based on this insight we develop an entire new hierarchy of optimized overlapping and nonoverlapping schwarz methods for maxwell s equations with greatly enhanced performance compared to the classical schwarz method we also derive for each algorithm asymptotic formulas for the optimized transmission conditions which can easily be used in implementations of the algorithms for problems with variable coefficients we illustrate our findings with numerical experiments

a maximum principle applied to quasi geostrophic equations we study the initial value problem for dissipative 2d quasi geostrophic equations proving local existence global results for small initial data in the super critical case decay of lp norms and asymptotic behavior of viscosity solution in the critical case our proofs are based on a maximum principle valid for more general flows

efficient large scale electromagnetic simulations using dynamically adapted meshes with the discontinuous galerkin method a framework for performing dynamic mesh adaptation with the discontinuous galerkin method dgm is presented adaptations include modifications of the local mesh step size h adaptation and the local degree of the approximating polynomials p adaptation as well as their combination the computation of the approximation within locally adapted elements is based on projections between finite element spaces fes which are shown to preserve an upper limit of the electromagnetic energy the formulation supports high level hanging nodes and applies precomputation of surface integrals for increasing computational efficiency error and smoothness estimates based on interface jumps are presented and applied to the fully hp adaptive simulation of two examples in one dimensional space a full wave simulation of electromagnetic scattering from a radar reflector demonstrates the applicability to large scale problems in three dimensional space

an implicit hybridized discontinuous galerkin method for the 3d time domain maxwell equations we present a time implicit hybridizable discontinuous galerkin hdg method for numerically solving the system of three dimensional 3d time domain maxwell equations this method can be seen as a fully implicit variant of classical so called dgtd discontinuous galerkin time domain methods that have been extensively studied during the last 10 years for the simulation of time domain electromagnetic wave propagation the proposed method has been implemented for dealing with general 3d problems discretized using unstructured tetrahedral meshes we provide numerical results aiming at assessing its numerical convergence properties by considering a model problem on one hand and its performance when applied to more realistic problems we also include some performance comparisons with a centered flux time implicit dgtd method

drift diffusion equations with fractional diffusion and the quasi geostrophic equation motivated by the critical dissipative quasi geostrophic equation we prove that drift diffusion equations with l 2 initial data and minimal assumptions on the drift are locally holder continuous as an application we show that solutions of the quasi geostrophic equation with initial l 2 data and critical diffusion 1 2 are locally smooth for any space dimension

lower bounds for an integral involving fractional laplacians and the generalized navier stokes equations in besov spaces when estimating solutions of dissipative partial differential equations in l p related spaces we often need lower bounds for an integral involving the dissipative term if the dissipative term is given by the usual laplacian lower bounds can be derived through integration by parts and embedding inequalities however when the laplacian is replaced by the fractional laplacian the approach of integration by parts no longer applies in this paper we obtain lower bounds for the integral involving by combining pointwise inequalities for with bernstein s inequalities for fractional derivatives as an application of these lower bounds we establish the existence and uniqueness of solutions to the generalized navier stokes equations in besov spaces the generalized navier stokes equations are the equations resulting from replacing in the navier stokes equations by

solution of the time harmonic maxwell equations using discontinuous galerkin methods we present numerical results concerning the solution of the time harmonic maxwell equations discretized by discontinuous galerkin methods in particular a numerical study of the convergence which compares different strategies proposed in the literature for the elliptic maxwell equations is performed in the two dimensional case

positive solutions for a class of singular dirichlet problems abstract we consider a dirichlet elliptic problem driven by the laplacian with singular and superlinear nonlinearities the singular term appears on the left hand side while the superlinear perturbation is parametric with parameter 0 and it need not satisfy the ar condition having as our starting point the work of diaz morel oswald 1987 3 we show that there is a critical parameter value such that for all the problem has two positive solutions while for there are no positive solutions what happens in the critical case is an interesting open problem

an explicit hybridizable discontinuous galerkin method for the acoustic wave equation abstract we present an explicit hybridizable discontinuous galerkin hdg method for numerically solving the acoustic wave equation the method is fully explicit high order accurate in both space and time and coincides with the classic discontinuous galerkin dg method with upwinding fluxes for a particular choice of its stabilization function this means that it has the same computational complexity as other explicit dg methods however just as its implicit version it provides optimal convergence of order k 1 for all the approximate variables including the gradient of the solution and when the time stepping method is of order k 2 it displays a superconvergence property which allow us by means of local postprocessing to obtain new improved approximations of the scalar field variables at any time levels for which an enhanced accuracy is required in particular the new approximations converge with order k 2 in the l 2 norm for k 1 these properties do not hold for all numerical fluxes indeed our results show that when the hdg numerical flux is replaced by the lax friedrichs flux the above mentioned superconvergence properties are lost although some are recovered when the lax friedrichs flux is used only in the interior of the domain finally we extend the explicit hdg method to treat the wave equation with perfectly matched layers we provide numerical examples to demonstrate the performance of the proposed method

the maximum principle and the global attractor for the dissipative 2d quasi geostrophic equations the long time behavior of the solutions to the two dimensional dissipative quasi geostrophic equations is studied we obtain a new positivity lemma which improves a previous version of a cordoba and d cordoba 10 and 11 as an application of the new positivity lemma we obtain the new maximum principle i e the decay of the solution in l p for any p 2 when f is zero as a second application of the new positivity lemma for the sub critical dissipative case with open image in new window the existence of the global attractor for the solutions in the space h s for any s 2 1 is proved for the case when the time independent f is non zero therefore the global attractor is infinitely smooth if f is this significantly improves the previous result of berselli 2 which proves the existence of an attractor in some weak sense for the case 1 the global attractor exists in h s for any s 0 and the estimate of the hausdorff and fractal dimensions of the global attractor is also available

global cauchy problem of 2d generalized magnetohydrodynamic equations abstract in this paper we prove the global in time existence of smooth solutions of the 2d generalized magnetohydrodynamic equations with zero viscosity when the power of the laplacian operator in the dissipation term of the induction equation is between 1 and 2 we also show a regularity criterion on the direction of the magnetic field b 0 b b when 1

existence and uniqueness results for the 2 d dissipative quasi geostrophic equation abstract this paper concerns itself with besov space solutions of the 2 d quasi geostrophic qg equation with dissipation induced by a fractional laplacian the goal is threefold first to extend a previous result on solutions in the inhomogeneous besov space b 2 q r j wu global solutions of the 2d dissipative quasi geostrophic equation in besov spaces siam j math anal 36 2004 2005 1014 1030 to cover the case when r 2 2 second to establish the global existence of solutions in the homogeneous besov space b p q r with general indices p and q and third to determine the uniqueness of solutions in any one of the four spaces b 2 q s b p q r l q 0 t b 2 q s 2 q and l q 0 t b p q r 2 q where s 2 2 and r 1 2 2 p

locally divergence free discontinuous galerkin methods for the maxwell equations in this paper we develop the locally divergence free discontinuous galerkin method for numerically solving the maxwell equations the distinctive feature of the method is the use of approximate solutions that are exactly divergence free inside each element as a consequence this method has a smaller computational cost than that of the discontinuous galerkin method with standard piecewise polynomial spaces we show that in spite of this fact it produces approximations of the same accuracy we also show that this method is more efficient than the discontinuous galerkin method using globally divergence free piecewise polynomial bases finally a post processing technique is used to recover 2k 1 th order of accuracy when piecewise polynomials of degree k are used

on the 2d critical and supercritical dissipative quasi geostrophic equation in besov spaces abstract we prove the local smoothing effect of the 2d critical and supercritical dissipative quasi geostrophic equations in critical besov spaces as an application a global well posedness result is established by adapting a method in kiselev nazarov and volberg 2007 16 and an idea in dong and du 2008 15 with suitable modifications moreover we show that the unique solution obtained in chen miao and zhang 2007 11 is a classical solution these generalize some previous results in dong 2010 13 dong and du 2008 15 the main ingredients of the proofs are two commutator estimates and the preservation of suitable modulus of continuity of the solutions

a reduced order dg formulation based on pod method for the time domain maxwell s equations in dispersive media abstract in this work a proper orthogonal decomposition pod method is applied to time domain maxwell s equations coupled to a drude dispersion model which are discretized in space by a discontinuous galerkin dg method an auxiliary differential equation ade method is used to represent the constitutive relation for the dispersive medium a pod dgtd formulation with lower dimension and sufficiently high accuracy is established together with the description of the pod reduced order basis its construction from a snapshot set and its application to the solution of the time domain maxwell s equations the overall goal is to reduce the computational time while maintaining an acceptable level of accuracy in order to obtain an efficient time domain solver to be used as a starting point for an optimization strategy we provide results from numerical experiments for two dimensional problems that illustrate the capabilities of the proposed pod dgtd formulation and assess its efficiency

remarks on the breakdown of smooth solutions for the 3 d euler equations the authors prove that the maximum norm of the vorticity controls the breakdown of smooth solutions of the 3 d euler equations in other words if a solution of the euler equations is initially smooth and loses its regularity at some later time then the maximum vorticity necessarily grows without bound as the critical time approaches equivalently if the vorticity remains bounded a smooth solution persists

a domain decomposition method for solving the three dimensional time harmonic maxwell equations discretized by discontinuous galerkin methods we present here a domain decomposition method for solving the three dimensional time harmonic maxwell equations discretized by a discontinuous galerkin method in order to allow the treatment of irregularly shaped geometries the discontinuous galerkin method is formulated on unstructured tetrahedral meshes the domain decomposition strategy takes the form of a schwarz type algorithm where a continuity condition on the incoming characteristic variables is imposed at the interfaces between neighboring subdomains a multifrontal sparse direct solver is used at the subdomain level the resulting domain decomposition strategy can be viewed as a hybrid iterative direct solution method for the large sparse and complex coefficients algebraic system resulting from the discretization of the time harmonic maxwell equations by a discontinuous galerkin method

spatial analyticity of the solutions to the subcritical dissipative quasi geostrophic equations we employ a new bilinear estimate to show that solutions to the subcritical dissipative quasi geostrophic equations with initial data in the scaling invariant lebesgue space are analytic in space variables some decay in time estimates for space time derivatives are also obtained