remarks on marti s method for solving first kind equations a convergence theorem for marti s method for solving the fredholm first kind equation kf g is corrected additional convergence results are given marti s method is shown to be well posed under perturbations in g and under certain types of perturbations in k

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

a multiscale discontinuous galerkin method for convection diffusion reaction problems we provide a general framework of multiscale discontinuous galerkin methods developed in buffa et al 2006 hughes et al 2006 for general second order partial differential equations we establish stability of the method and prove the error estimates

coupling discontinuous galerkin discretizations using mortar finite elements for advection diffusion reaction problems we investigate dg dg domain decomposition coupling using mortar finite elements to approximate the solution to general second order partial differential equations we weakly impose an inflow boundary condition on the inflow part of the interface and the dirichlet boundary condition on the elliptic part of the interface via lagrange multipliers we obtain the matching condition by imposing the continuity of the total flux through the interface and the continuity of the solution on the elliptic parts of the interface the diffusion coefficient is allowed to be degenerate and the mortar interface couples efficiently the multiphysics problems the discrete problem is solvable in each subdomain in terms of lagrange multipliers and the resulting algorithm is easily parallelizable the subdomain grids need not match and the mortar grid may be much coarser giving a two scale method convergence results in terms of the fine subdomain scale h and the coarse mortar scale h are then established a non overlapping parallelizable domain decomposition algorithm arbogast et al 2007 reduces the coupled system to an interface mortar problem the properties of the interface operator are discussed

two level mortar domain decomposition preconditioners for heterogeneous elliptic problems abstract we consider a second order elliptic problem with a heterogeneous coefficient which models for example single phase flow through a porous medium we write this problem in mixed form and approximate it for parallel computation using the multiscale mortar domain decomposition mixed finite element method which gives rise to a saddle point linear system we use a relatively fine mortar space which allows us to enforce continuity of the normal velocity flux or nearly so in the case of nonmatching meshes to solve the schur complement linear system for the mortar unknowns we propose a two level preconditioner based on the interfaces between subdomains the coarse preconditioner also uses the multiscale mortar domain decomposition method but with instead a very coarse mortar space we show that the prolongation operator of the coarse mortar to the fine is defined uniquely by the condition that the l 2 projection of a coarse mortar agrees with its projection onto the space of normal velocity fluxes i e no energy is introduced when changing mortar scales the local smoothing preconditioner is based on block jacobi using blocks defined by the interfaces we use restrictive smoothing domains that are smaller normal to the interfaces and overlapping in the directions tangential to the interfaces in the simplest case the condition number of the preconditioned interface operator is bounded by a multiple of log 1 h h 2 we show several numerical examples involving strongly heterogeneous porous media to demonstrate the efficiency and robustness of the preconditioner we see that it is often desirable and sometimes necessary to use a piecewise linear or higher order coarse mortar space to achieve good convergence for heterogeneous problems

a hybrid discontinuous galerkin method for elliptic problems a new family of hybrid discontinuous galerkin methods is studied for second order elliptic equations our proposed method is a generalization of the cell boundary element cbe method y jeon and e j park appl numer math 58 2008 pp 800 814 which allows high order polynomial approximations our method can be viewed as a hybridizable discontinuous galerkin method b cockburn j gopalakrishnan and r lazarov siam j numer anal 47 2009 pp 1319 1365 using a bauman oden type local solver the method conserves the mass in each element and the average flux is continuous across the interelement boundary for even degree polynomial approximations optimal order error estimates measured in the energy norm are proved numerical examples are presented to show the performance of the method

a multiscale discontinuous galerkin method with the computational structure of a continuous galerkin method proliferation of degrees of freedom has plagued discontinuous galerkin methodology from its inception over 30 years ago this paper develops a new computational formulation that combines the advantages of discontinuous galerkin methods with the data structure of their continuous galerkin counterparts the new method uses local element wise problems to project a continuous finite element space into a given discontinuous space and then applies a discontinuous galerkin formulation the projection leads to parameterization of the discontinuous degrees of freedom by their continuous counterparts and has a variational multiscale interpretation this significantly reduces the computational burden and at the same time little or no degradation of the solution occurs in fact the new method produces improved solutions compared with the traditional discontinuous galerkin method in some situations

a hybridized finite element method for the stokes problem a locally conservative hybridized finite element method for stokes equations is presented and analyzed the hybridized approach reduces a lot of degrees of freedom especially for pressure approximation in our approach the pressure is determined locally up to a constant therefore the global stiffness system contains only the average of pressure variable on each cell as unknowns

high order discontinuous galerkin methods with lagrange multiplier for hyperbolic systems of conservation laws in this work we present novel high order discontinuous galerkin methods with lagrange multiplier dglm for hyperbolic systems of conservation laws lagrange multipliers are introduced on the inter element boundaries via the concept of weak divergence static condensation on element unknowns considerably reduces globally coupled degrees of freedom resulting in the stiffness equations in the lagrange multipliers only we first establish stability results and provide conditions on the stabilization parameter which plays an important role in resolving discontinuities as well accuracy tests are then performed which shows optimal convergence in the l2 norm extensive numerical results indicate that the dglm has potentials in delivering high order accurate information for various problems in hyperbolic conservation laws numerical examples include inviscid burgers equations shallow water equations subcritical flow and supercritical upstream subcritical downstream flow and 2d circular dam break and compressible euler equations intersection of mach 3 and sods shock tube

new locally conservative finite element methods on a rectangular mesh a new family of locally conservative finite element methods for a rectangular mesh is introduced to solve second order elliptic equations our approach is composed of generating pde adapted local basis and solving a global matrix system arising from a flux continuity equation quadratic and cubic elements are analyzed and optimal order error estimates measured in the energy norm are provided for elliptic equations next this approach is exploited to approximate stokes equations numerical results are presented for various examples including the lid driven cavity problem

high order dg dglm method for hyperbolic conservation laws abstract a high order discontinuous galerkin method in time combined with discontinuous galerkin method with lagrange multiplier dglm kim 2015 4 in space is proposed to approximate the solution to hyperbolic conservation laws with boundary conditions stability of the approximate solution is proved in a broken l 2 l 2 norm and also in an l l 2 norm error estimates of o h r 1 2 k n q 1 2 with p r e and p q j n elements r q d 1 2 are derived in a broken l 2 l 2 norm where h and k n are the maximum diameters of the elements and the time step of j n respectively j n is the time interval and d is the dimension of the spatial domain an explanation on algorithmic aspects is given

analysis of adaptive mesh refinement for imex discontinuous galerkin solutions of the compressible euler equations with application to atmospheric simulations the resolutions of interests in atmospheric simulations require prohibitively large computational resources adaptive mesh refinement amr tries to mitigate this problem by putting high resolution in crucial areas of the domain we investigate the performance of a tree based amr algorithm for the high order discontinuous galerkin method on quadrilateral grids with non conforming elements we perform a detailed analysis of the cost of amr by comparing this to uniform reference simulations of two standard atmospheric test cases density current and rising thermal bubble the analysis shows up to 15 times speed up of the amr simulations with the cost of mesh adaptation below 1 of the total runtime we pay particular attention to the implicit explicit imex time integration methods and show that the ark2 method is more robust with respect to dynamically adapting meshes than bdf2 preliminary analysis of preconditioning reveals that it can be an important factor in the amr overhead the compiler optimizations provide significant runtime reduction and positively affect the effectiveness of amr allowing for speed ups greater than it would follow from the simple performance model

an implicit high order hybridizable discontinuous galerkin method for nonlinear convection diffusion equations in this paper we present hybridizable discontinuous galerkin methods for the numerical solution of steady and time dependent nonlinear convection diffusion equations the methods are devised by expressing the approximate scalar variable and corresponding flux in terms of an approximate trace of the scalar variable and then explicitly enforcing the jump condition of the numerical fluxes across the element boundary applying the newton raphson procedure and the hybridization technique we obtain a global equation system solely in terms of the approximate trace of the scalar variable at every newton iteration the high number of globally coupled degrees of freedom in the discontinuous galerkin approximation is therefore significantly reduced we then extend the method to time dependent problems by approximating the time derivative by means of backward difference formulae when the time marching method is p 1 th order accurate and when polynomials of degree p 0 are used to represent the scalar variable each component of the flux and the approximate trace we observe that the approximations for the scalar variable and the flux converge with the optimal order of p 1 in the l 2 norm finally we apply element by element postprocessing schemes to obtain new approximations of the flux and the scalar variable the new approximate flux which has a continuous interelement normal component is shown to converge with order p 1 in the l 2 norm the new approximate scalar variable is shown to converge with order p 2 in the l 2 norm the postprocessing is performed at the element level and is thus much less expensive than the solution procedure for the time dependent case the postprocessing does not need to be applied at each time step but only at the times for which an enhanced solution is required extensive numerical results are provided to demonstrate the performance of the present method

region of attraction analysis of nonlinear stochastic systems using polynomial chaos expansion abstract a method is presented to estimate the region of attraction roa of stochastic systems with finite second moment and uncertainty dependent equilibria the approach employs polynomial chaos pc expansions to represent the stochastic system by a higher dimensional set of deterministic equations we first show how the equilibrium point of the deterministic formulation provides the stochastic moments of an uncertainty dependent equilibrium point of the stochastic system a connection between the boundedness of the moments of the stochastic system and the lyapunov stability of its pc expansion is then derived defining corresponding notions of a roa for both system representations we show how this connection can be leveraged to recover an estimate of the roa of the stochastic system from the roa of the pc expanded system two optimization programs obtained from sum of squares programming techniques are provided to compute inner estimates of the roa the first optimization program uses the lyapunov stability arguments to return an estimate of the roa of the pc expansion based on this result and user specifications on the moments for the initial conditions the second one employs the shown connection to provide the corresponding roa of the stochastic system the method is demonstrated by two examples

a fully implicit domain decomposition algorithm for shallow water equations on the cubed sphere popular approaches for solving shallow water equations swes for climate modeling are explicit and semiimplicit methods and both have certain constraints on the time step size in this paper we propose and study a fully implicit method which imposes no limit on the time step size but requires the solution of a large sparse nonlinear system of equations at every time step the focus of the paper is a parallel fully coupled newton krylov ras algorithm with a jacobian matrix explicitly calculated on a weakly nonmatching cubed sphere mesh here ras is a restricted additive schwarz method we show numerically that with such a preconditioned nonlinearly implicit method the time step size is no longer constrained by the cfl condition and we report superlinear speedup of the algorithm on machines with thousands of processors for problems with smooth and nonsmooth solutions

coupling discontinuous galerkin and mixed finite element discretizations using mortar finite elements discontinuous galerkin dg and mixed finite element mfe methods are two popular methods that possess local mass conservation in this paper we investigate dg dg and dg mfe domain decomposition couplings using mortar finite elements to impose weak continuity of fluxes and pressures on the interface the subdomain grids need not match and the mortar grid may be much coarser giving a two scale method convergence results in terms of the fine subdomain scale h and the coarse mortar scale h are established for both types of couplings in addition a nonoverlapping parallel domain decomposition algorithm is developed which reduces the coupled system to an interface mortar problem the properties of the interface operator are analyzed

a discontinuous galerkin method with lagrange multiplier for hyperbolic conservation laws with boundary conditions we introduce a discontinuous galerkin method with lagrange multiplier dglm to approximate the solution to the hyperbolic conservation laws with boundary conditions lagrange multipliers are introduced on the edge face of the element via weak divergence wang and ye 2014 the final global system has reduced numbers of unknowns of the standard dg methods numerical fluxes from finite volume difference method are not considered for the time discretization backward euler difference method is used stability of the approximate solution is proved in energy norm discontinuity of the solution is allowed in the error analysis local error estimates of o hr 12 t with pr e elements rd 12 are derived where h and t are the maximum diameter of the elements and time steps respectively and d is the dimension of the spatial domain the high order approximation is obtained under an appropriate condition on the stabilizing parameter it is shown that the method preserves the property of the local mass conservation an explanation on algorithmic aspects is given

a weak galerkin mixed finite element method for second order elliptic problems abstract a new weak galerkin wg method is introduced and analyzed for the second order elliptic equation formulated as a system of two first order linear equations this method called wg mfem is designed by using discontinuous piecewise polynomials on finite element partitions with arbitrary shape of polygons polyhedra the wg mfem is capable of providing very accurate numerical approximations for both the primary and flux variables allowing the use of discontinuous approximating functions on arbitrary shape of polygons polyhedra makes the method highly flexible in practical computation optimal order error estimates in both discrete h and l norms are established for the corresponding weak galerkin mixed finite element solutions

analysis of a multiscale discontinuous galerkin method for convection diffusion problems we study a multiscale discontinuous galerkin method introduced in t j r hughes g scovazzi p bochev and a buffa comput meth appl mech engrg 195 2006 pp 2761 2787 that reduces the computational complexity of the discontinuous galerkin method seemingly without adversely affecting the quality of results for a stabilized variant we are able to obtain the same error estimates for the convection diffusion equation as for the usual discontinuous galerkin method we assess the stability of the unstabilized case numerically and find that the inf sup constant is positive bounded uniformly away from zero and very similar to that for the usual discontinuous galerkin method

a simple illustration of a weak spectral cascade the textbook first encounter with nonlinearity in a partial differential equation pde is the first order wave equation u t u u x 0 often referred to as the inviscid burgers equation this equation is familiar to many in the theoretical contexts of characteristics wavebreaking or shock propagation another canonical behavior contained within this simplest of pdes is the spectral cascade surprisingly buried in a little known 1964 article by g w platzman is an elegant example of an exact fourier series solution associated with a purely sinusoidal initial condition this fourier representation valid prior to wavebreaking is generalized to arbitrary continuous initial conditions on both the periodic and infinite domains for the specific example of platzman s original problem the fourier coefficients decay exponentially with increasing wavenumber and the decay rate flattens to zero precisely at the time of wavebreaking it is demonstrated that two simplified descriptions a downscale truncat