on the balancing principle for some problems of numerical analysis we discuss a choice of weight in penalization methods the motivation for the use of penalization in computational mathematics is to improve the conditioning of the numerical solution one example of such improvement is a regularization where a penalization substitutes an ill posed problem for a well posed one in modern numerical methods for pdes a penalization is used for example to enforce a continuity of an approximate solution on non matching grids a choice of penalty weight should provide a balance between error components related with convergence and stability which are usually unknown in this paper we propose and analyze a simple adaptive strategy for the choice of penalty weight which does not rely on a priori estimates of above mentioned components it is shown that under natural assumptions the accuracy provided by our adaptive strategy is worse only by a constant factor than one could achieve in the case of known stability and convergence rates finally we successfully apply our strategy for self regularization of volterra type severely ill posed problems such as the sideways heat equation and for the choice of a weight in interior penalty discontinuous approximation on non matching grids numerical experiments on a series of model problems support theoretical results

analysis of semi implicit dgfem for nonlinear convection diffusion problems on nonconforming meshes the paper deals with the numerical analysis of a scalar nonstationary nonlinear convection diffusion equation the space discretization is carried out by the discontinuous galerkin finite element method dgfem on general nonconforming meshes formed by possibly nonconvex elements with nonsymmetric treatment of stabilization terms and interior and boundary penalty the time discretization is carried out by a semi implicit euler scheme in which the diffusion and stabilization terms are treated implicitly whereas the nonlinear convective terms are treated explicitly we derive a priori asymptotic error estimates in the discrete l l2 l l2 norm l2 h1 l2 h1 seminorm and l h1 l h1 seminorm with respect to the mesh size h and time step numerical examples demonstrate the accuracy of the method and manifest the effect of nonconvexity of elements and nonconformity of the mesh

discontinuous galerkin least squares finite element methods for singularly perturbed reaction diffusion problems with discontinuous coefficients and boundary singularities in this paper a discontinuous galerkin least squares finite element method is developed for singularly perturbed reaction diffusion problems with discontinuous coefficients and boundary singularities by recasting the second order elliptic equations as a system of first order equations in a companion paper lin in siam j numer anal 47 89 108 2008 a similar method has been developed for problems with continuous data and shown to be well posed uniformly convergent and optimal in convergence rate in this paper the method is modified to take care of conditions that arise at interfaces and boundary singularities coercivity and uniform error estimates for the finite element approximation are established in an appropriately scaled norm numerical examples confirm the theoretical results

a discontinuous hp finite element method for diffusion problems 1 d analysis abstract this paper presents the mathematical analysis of a new variant of the discontinuous galerkin method which is applicable to the numerical solution of diffusion problems not requiring auxiliary variables such as those used in mixed methods the focus of this study is on a class of linear second order boundary value problems for which we prove stability and a priori error estimates in both the finite and infinite dimensional spaces

consistent discontinuous finite elements in elastodynamics finite element discontinuities with respect to time have recently been extremely used in elastodynamic problems due to their natural utilization in combination with adaptive methods and their efficiency in discontinuity capturing techniques for non smooth problems in this work we present some theoretical aspects and numerical results concerning the use of spatial discontinuities in a consistent finite element method for the same class of problems we first review some formulations for the elastostatic problem and prove two korn like inequalities which are very useful for the derivation of convergence rates in sobolev norms next we present formulations for the dynamic case along with comments on their properties and estimates of convergence rates for smooth solutions followed by numerical investigations of a typically non smooth problem involving classical and emerging variational formulations we also show some numerical experiments with finite element spaces enriched by discontinuous functions other than piecewise lagrangian polynomials

analysis of a bdf dgfe scheme for nonlinear convection diffusion problems we deal with the numerical solution of a scalar nonstationary nonlinear convection diffusion equation we employ a combination of the discontinuous galerkin finite element method for the space semi discretization and the k step backward difference formula for the time discretization the diffusive and stabilization terms are treated implicitly whereas the nonlinear convective term is treated by a higher order explicit extrapolation method which leads to the necessity to solve only a linear algebraic problem at each time step we analyse this scheme and derive a priori asymptotic error estimates in the discrete l l 2 norm and the l 2 h 1 seminorm with respect to the mesh size h and time step for k 2 3 numerical examples verifying the theoretical results are presented

weak galerkin methods for time dependent maxwell s equations abstract this paper adapts the weak galerkin wg finite element scheme to maxwell s equations in the time domain developed by wang and ye in 2011 the wg scheme is a discontinuous galerkin like method that relies on a new notion of the discrete weak derivative the wg method as applied to maxwell s equations uses a discrete weak curl and an extra stabilization term to enforce a weak continuity of the solution across elements semi discrete and fully discrete schemes are developed and are shown to be unconditionally stable an optimal order of convergence is proven in an appropriate energy norm and confirmed by numerical results for clarity the theoretical analysis is carried out for 3 d case some details showing how to implement this wg scheme in 2 d are provided and numerical results are given in 2 d

augmented high order finite volume element method for elliptic pdes in non smooth domains convergence study abstract the accuracy of a finite element numerical approximation of the solution of a partial differential equation can be spoiled significantly by singularities this phenomenon is especially critical for high order methods in this paper we show that if the pde is linear and the singular basis functions are homogeneous solutions of the pde the augmentation of the trial function space for the finite volume element method fvem can be done significantly simpler than for the finite element method when the trial function space is augmented for the fvem all the entries in the matrix originating from the singular basis functions in the discrete form of the pde are zero and the singular basis functions only appear in the boundary conditions that is to say there is no need to integrate the singular basis functions over the elements and the sparsity of the matrix is preserved without special care fvem numerical convergence studies on two dimensional triangular grids are presented using basis functions of arbitrary high order confirming the same order of convergence for singular solutions as for smooth solutions

godunov mixed methods for advection diffusion equations in multidimensions time split methods for multidimensional advection diffusion equations are considered in these methods advection is approximated by a godunov type procedure and diffusion is approximated by a low

a new posteriori error estimates for a discontinuous galerkin method for a reaction diffusion problem in this paper a new posteriori error estimates and an error indicator for locally mass conservative methods for subsurface flow are given the methods are based on discontinuous approximation spaces and the penalty terms on the normal fluxes across the element interfaces are added to the bilinear form to obtain the nonsymmetric interior penalty discontinuous galerkin methods the global explicit estimates of the error in the l 2 norm are derived and an implicit indicator of the error in the energy norm is numerically studied

performance of discontinuous galerkin methods for elliptic pdes in this paper we compare the performance of several discontinuous galerkin dg methods for elliptic partial differential equations pdes on a model problem theoretical estimates of the condition number of the stiffness matrix are given for dg methods whose bilinear form is symmetric and which are shown to be numerically sharp then the efficiency of the methods in the computation of both the potential and its gradient is tested on unstructured triangular meshes

a discontinuous finite element formulation for helmholtz equation abstract a discontinuous finite element formulation for helmholtz problems is presented with c 0 continuity across inter element boundaries enforced in a weak sense depending on two parameters and whose choice is crucial for the performance of the proposed method these parameters are chosen through one dimension numerical experiments aiming at minimizing the pollution error optimal values of these parameters are then adopted in more general situations the accuracy and stability of the proposed formulation for linear and bilinear shape functions are demonstrated in several numerical examples in one and two dimensions

hp version interior penalty dgfems for the biharmonic equation we construct hp version interior penalty discontinuous galerkin finite element methods dgfems for the biharmonic equation including symmetric and nonsymmetric interior penalty discontinuous galerkin methods and their combinations semisymmetric methods our main concern is to establish the stability and to develop the a priori error analysis of these methods we establish error bounds that are optimal in h and slightly suboptimal in p the theoretical results are confirmed by numerical experiments

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

statistical properties and decay of correlations for interval maps with critical points and singularities we consider a class of piecewise smooth one dimensional maps with critical points and singularities possibly with infinite derivative under mild summability conditions on the growth of the derivative on critical orbits we prove the central limit theorem and a vector valued almost sure invariance principle we also obtain results on decay of correlations and large deviations

discontinuous finite element formulations applied to cracked elastic domains the solutions of boundary value problems defined on cracked domains are usually non smooth in the surroundings of the crack in this work we formulate the elasticity problem of a body with such geometric characteristic in a number of equivalent variational alternatives and show that we can take advantage of the theory of discontinuous finite elements in order to approximate its solution in an interesting way at little higher programming cost in comparison with the classical galerkin method the idea consists in splitting the global domain into a number of regions in which local mesh refinements are undertaken independently producing irregular meshes with non matching elements that are suitable to be used in discontinuous finite element methods this strategy seems to be attractive to be employed in situations that we know in advance where the critical regions of the domain are located as well as in adaptive techniques

a discontinuous hp finite element method for convection diffusion problems this paper presents a new method which exhibits the best features of both finite volume and finite element techniques special attention is given to the issues of conservation flexible accuracy and stability the method is elementwise conservative the order of polynomial approximation can be adjusted element by element and the stability is not based on the introduction of artificial diffusion but on the use of a very particular finite element formulation with discontinuous basis functions the method supports h p and hp approximations and can be applied to any type of meshes including non matching grids a priori error estimates and numerical experiments on representative model problems indicate that the method is robust and capable of delivering high accuracy

analysis of a combined cg1 dg2 method for the transport equation in this paper we introduce a reduced discontinuous galerkin method in which the space of continuous piecewise linear functions cg1 is enriched with discontinuous piecewise quadratics dg2 the resultant finite element approximation is continuous at the vertices of the mesh and discontinuous across edges faces we analyze the properties of the cg1 dg2 discretization in the context of a steady linear transport equation the presented a priori error estimate shows that the discontinuous enrichment stabilizes the continuous coarse scale component and delivers optimal convergence rates numerical studies for steady and unsteady convection problems confirm this result

discontinuous galerkin finite element in time for solving periodic differential equations the dynamics of rotating systems can be modeled by ordinary differential equations with periodic time varying coefficients and forcing discontinuous galerkin dg methods which are known to be higher order accurate a stable and locally conservative are used to solve such equations in this work dg methods are shown to find the steady state periodic solution of a constant coefficient structural dynamics problem with periodic forcing and b helicopter rotor rigid blade flapping equation which is a parametric ordinary differential equation with periodic time varying coefficients and periodic forcing convergence of the numerical solution obtained is studied for both the above equations while h and p are refined separately it is found that the dg method works well for these problems and allows the development of an element level error estimate which can be used to monitor convergence and hence develop adaptive schemes it is clear from the results that even though higher order polynomials result in much faster convergence with reduced computational effort it also results in deterioration of the numerical performance for p 12 two adaptive strategies based on the element level displacement jump norm i e h and hp refinement are proposed and demonstrated on problems with higher harmonic forcing primary controls and parametric variations

analysis of space time discontinuous galerkin method for nonlinear convection diffusion problems the paper presents the theory of the discontinuous galerkin finite element method for the space time discretization of a nonstationary convection diffusion initial boundary value problem with nonlinear convection and linear diffusion the problem is not singularly perturbed with dominating convection the discontinuous galerkin method is applied separately in space and time using in general different space grids on different time levels and different polynomial degrees p and q in space and time dicretization in the space discretization the nonsymmetric symmetric and incomplete interior and boundary penalty nipg sipg iipg approximation of diffusion terms is used the paper is concerned with the proof of error estimates in l 2 l 2 and dg norm formed by the l 2 h 1 seminorm and penalty terms a special technique based on the use of the gauss radau interpolation and numerical integration has been used for the derivation of an abstract error estimate in the dg norm the error estimates are optimal with respect to the size of the space grid they are optimal with respect to the time step if the dirichlet boundary condition has behaviour in time as a polynomial of degree q

analysis of a family of discontinuous galerkin methods for elliptic problems the one dimensional case in this paper we analyze a family of discontinuous galerkin methods parameterized by two real parameters for elliptic problems in one dimension our main results are 1 a complete inf sup stability analysis characterizing the parameter values yielding a stable scheme and energy norm error estimates as a direct consequence thereof 2 an analysis of the error in l2 where the standard duality argument only works for special parameter values yielding a symmetric bilinear form and different orders of convergence are obtained for odd and even order polynomials in the nonsymmetric case the analysis is consistent with numerical results and similar behavior is observed in two dimensions

an h1 ph coercive discontinuous galerkin formulation for the poisson problem 1 d analysis coercivity of the bilinear form in a continuum variational problem is a fundamental property for finite element discretizations by the classical lax milgram theorem any conforming discretization of a coercive variational problem is stable i e discrete approximations are well posed and possess unique solutions irrespective of the specifics of the underlying approximation space based on the prototypical one dimensional poisson problem we establish in this work that most concurrent discontinuous galerkin formulations for second order elliptic problems represent instances of a generic conventional formulation and that this generic formulation is noncoercive consequently all conventional discontinuous galerkin formulations are a fortiori noncoercive and typically their well posedness is contingent on approximation space dependent stabilization parameters moreover we present a new symmetric nonconventional discontinuous galerkin formulation based on element green s functions and the data local to the edges we show that the new discontinuous galerkin formulation is coercive on the broken sobolev space h1 ph viz the space of functions that are elementwise in the h1 sobolev space the coercivity of the new formulation is supported by calculations of discrete inf sup constants and numerical results are presented to illustrate the optimal convergence behavior in the energy norm and in the l2 norm

hp discontinuous galerkin methods for strongly nonlinear elliptic boundary value problems in this paper we have analyzed a one parameter family of hp discontinuous galerkin methods for strongly nonlinear elliptic boundary value problems nabla cdot rm a u nabla u f u nabla u 0 with dirichlet boundary conditions these methods depend on the values of the parameter theta in 1 1 where 1 corresponds to the nonsymmetric and 1 corresponds to the symmetric interior penalty methods when rm a u nabla u nabla u and f u u f that is for the poisson problem the error estimate in the broken h 1 norm which is optimal in h mesh size and suboptimal in p degree of approximation is derived using piecewise polynomials of degree p 2 when the solution u in h 5 2 omega in the case of linear elliptic problems also this estimate is optimal in h and suboptimal in p further optimal error estimate in the l 2 norm when 1 is derived numerical experiments are presented to illustrate the theoretical results

a discontinuous finite element based domain decomposition method in this work we introduce a somewhat unconventional finite element based nonoverlapping domain decomposition method the convenient space decomposition method csd for parallel computers that can be applied to a wide class of boundary value problems and which can only be implemented efficiently exclusively with primal variables by means of the use of discontinuous elements interesting aspects and properties are pointed out along with proofs for the most relevant results numerical experiments for purely diffusion problems are also presented in order to evaluate the effectiveness of the proposed scheme and to confirm the validity of the related theoretical results

a w cycle algorithm for a weakly over penalized interior penalty method we study a w cycle multigrid algorithm for a weakly over penalized interior penalty method for the two dimensional poisson problem on convex polygonal domains

a discontinuoushpfinite element method for diffusion problems we present an extension of the discontinuous galerkin method which is applicable to the numerical solution of diffusion problems the method involves a weak imposition of continuity conditions on the solution values and on fluxes across interelement boundaries within each element arbitrary spectral approximations can be constructed with different orderspin each element we demonstrate that the method is elementwise conservative a property uncharacteristic of high order finite elements for clarity we focus on a model class of linear second order boundary value problems and we developpriorierror estimates convergence proofs and stability estimates the results of numerical experiments onh andp convergence rates for representative two dimensional problems suggest that the method is robust and capable of delivering exponential rates of convergence

a discontinuous enrichment method for variable coefficient advection diffusion at high peclet number a discontinuous galerkin method with lagrange multipliers is presented for the solution of variable coefficient advection diffusion problems at high peclet number in this method the standard finite element polynomial approximation is enriched within each element with free space solutions of a local constant coefficient homogeneous counterpart of the governing partial differential equation hence in the two dimensional case the enrichment functions are exponentials each exhibiting a sharp gradient in a carefully chosen flow direction the continuity of the enriched approximation across the element interfaces is enforced weakly by the aforementioned lagrange multipliers numerical results obtained for two benchmark problems demonstrate that elements based on the proposed discretization method are far more competitive for variable coefficient advection diffusion analysis in the high peclet number regime than their standard galerkin and stabilized finite element comparables copyright 2010 john wiley sons ltd

interior penalty discontinuous approximations of convection diffusion problems with parabolic layers a nonsymmetric discontinuous galerkin finite element method with interior penalties is considered for two dimensional convection diffusion problems with regular and parabolic layers on an anisotropic shishkin type mesh with bilinear elements we prove error estimates uniformly in the perturbation parameter in an integral norm associated with this method on different types of interelement edges we derive the values of discontinuity penalization parameters numerical experiments complement the theoretical results

a characteristics mixed finite element method for advection dominated transport problems we define a new finite element method called the characteristics mixed method for approximating the solution to an advection dominated transport problem the method is based on a space time variational form of the advection diffusion equation our test functions are piecewise constant in space and in time they approximately follow the characteristics of the advective i e hyperbolic part of the equation thus the scheme uses a characteristic approximation to handle advection in time this is combined with a low order mixed finite element spatial approximation of the equation boundary conditions are incorporated in a natural and mass conservative fashion the scheme is completely locally conservative in fact on the discrete level fluid is transported along the approximate characteristics a postprocessing step is included in the scheme in which the approximation to the scalar unknown is improved by utilizing the approximate vector flux this has the effect of improving the rate of convergence of

continuous discontinuous finite element approximations of fourth order elliptic problems in structural and continuum mechanics with applications to thin beams and plates and strain gradient elasticity abstract a new finite element method for fourth order elliptic partial differential equations is presented and applied to thin bending theory problems in structural mechanics and to a strain gradient theory problem the method combines concepts from the continuous galerkin cg method the discontinuous galerkin dg method and stabilization techniques a brief review of the cg method the dg method and stabilization techniques highlights the advantages and disadvantages of these methods and suggests a new approach for the solution of fourth order elliptic problems a continuous discontinuous galerkin c dg method is proposed which uses c 0 continuous interpolation functions and is formulated in the primary variable only the advantage of this formulation over a more traditional mixed approach is that the introduction of additional unknowns and related difficulties can be avoided in the context of thin bending theory the c dg method leads to a formulation where displacements are the only degrees of freedom and no rotational degrees of freedom need to be considered the main feature of the c dg method is the weak enforcement of continuity of first and higher order derivatives through stabilizing terms on interior boundaries consistency stability and convergence of the method are shown analytically numerical experiments verify the theoretical results and applications are presented for bernoulli euler beam bending poisson kirchhoff plate bending and a shear layer problem using toupin mindlin strain gradient theory

discontinuous galerkin finite element differential calculus and applications to numerical solutions of linear and nonlinear partial differential equations this paper develops a discontinuous galerkin dg finite element differential calculus theory for approximating weak derivatives of sobolev functions and piecewise sobolev functions by introducing numerical one sided derivatives as building blocks various first and second order numerical operators such as the gradient divergence hessian and laplacian operator are defined and their corresponding calculus rules are established among the calculus rules are product and chain rules integration by parts formulas and the divergence theorem approximation properties and the relationship between the proposed dg finite element numerical derivatives and some well known finite difference numerical derivative formulas on cartesian grids are also established besides independent interest in numerical differentiation the primary motivation and goal of developing the dg finite element differential calculus is to solve partial differential equations it is shown that several existing finite element finite difference and discontinuous galerkin methods can be rewritten compactly using the proposed dg finite element differential calculus framework moreover new discontinuous galerkin methods for linear and nonlinear pdes are also obtained from the framework

stencil reduction algorithms for the local discontinuous galerkin method the problem of reducing the stencil of the local discontinuous galerkin method applied to second order differential operator is discussed heuristic algorithms to minimize the total number of non zero blocks of the reduced stiffness matrix are presented and tested on a wide variety of unstructured and structured grids in 2d and 3d copyright 2009 john wiley sons ltd

the discontinuous enrichment method we propose a finite element based discretization method in which the standard polynomial field is enriched within each element by a nonconforming field that is added to it the enrichment contains free space solutions of the homogeneous differential equation that are not represented by the underlying polynomial field continuity of the enrichment across element interfaces is enforced weakly by lagrange multipliers in this manner we expect to attain high coarse mesh accuracy without significant degradation of conditioning assuring good performance of the computation at any mesh resolution examples of application to acoustics and advection diffusion are presented

error estimates of b spline based finite element methods for the stationary quasi geostrophic equations of the ocean abstract this paper presents theoretical error estimates of b spline based finite element methods for the streamfunction formulation of the stationary quasi geostrophic equations which describe the large scale wind driven ocean circulation we introduce variational formulations of the streamfunction formulation inspired by the interior penalty discontinuous galerkin method dirichlet boundary conditions are weakly enforced in the formulations and stabilizations are achieved via nitsche s method existence and uniqueness of the approximation are proved and optimal error estimates in the energy norm are demonstrated under a small data assumption numerical experiments are performed to verify the theoretical error estimates on rectangular and l shape geometries

the local discontinuous galerkin method for the oseen equations we introduce and analyze the local discontinuous galerkin method for the oseen equations of incompressible fluid flow for a class of shape regular meshes with hanging nodes we derive optimal a priori estimates for the errors in the velocity and the pressure in l 2 and negative order norms numerical experiments are presented which verify these theoretical results and show that the method performs well for a wide range of reynolds numbers

discontinuous galerkin and mimetic finite difference methods for coupled stokes darcy flows on polygonal and polyhedral grids we study locally mass conservative approximations of coupled darcy and stokes flows on polygonal and polyhedral meshes the discontinuous galerkin dg finite element method is used in the stokes region and the mimetic finite difference method is used in the darcy region dg finite element spaces are defined on polygonal and polyhedral grids by introducing lifting operators mapping mimetic degrees of freedom to functional spaces optimal convergence estimates for the numerical scheme are derived results from computational experiments supporting the theory are presented

fluid structure interaction by the discontinuous galerkin method for large deformations a discontinuous galerkin method for large deformation fluid structure interaction problems is developed the fluid structure interface can be arbitrarily aligned relative to the fluid grid an eulerian description is used for the fluid with a lagrangian description of the solid results are presented for several examples that show excellent agreement with immersed fictitious element methods copyright 2008 john wiley sons ltd

hierarchical high order finite element spaces in h div ω h1 ω for a stabilized mixed formulation of darcy problem abstract the classical dual mixed finite element method for flow simulations is based on h div conforming approximation spaces for the flux which guarantees continuous normal components on element interfaces and discontinuous approximations in l 2 for the pressure however stability and convergence can only be obtained for compatible approximation spaces stabilized finite element methods may provide an alternative stable procedure to avoid this kind of delicate balance the main purpose of this paper is to present a high order finite element methodology to solve the darcy problem based on the combination of an unconditionally stable mixed finite element method with a hierarchical methodology for the construction of finite dimensional subspaces of h div and h 1 the chosen stabilized method is free of mesh dependent stabilization parameters and allows for the use of different high order finite element approximations for the flux and the pressure variables without requiring any compatibility constraint as required in mixed methods for these problems convergence studies are presented comparing the numerical solutions obtained for different approximation orders on quadrilateral elements with the ones given by classical mixed formulation with raviart thomas elements

bubble stabilized discontinuous galerkin method for parabolic and elliptic problems in this paper we give an analysis of a bubble stabilized discontinuous galerkin method for elliptic and parabolic problems the method consists of stabilizing the numerical scheme by enriching the discontinuous affine finite element space elementwise by quadratic bubbles this approach leads to optimal convergence in the space and time discretization parameters

analysis of a nonsymmetric discontinuous galerkin method for elliptic problems stability and energy error estimates in this paper we analyze a nonsymmetric discontinuous galerkin method for elliptic problems proposed by oden babuska and baumann our main results are a complete inf sup stability analysis and as a consequence error estimates in a mesh dependent energy norm allowing variable meshsize and order of polynomials the analysis is carried out in two spatial dimensions on an unstructured triangulation

an estimate of the babuska brezzi inf sup discrete stability constant for general linear petrov galerkin finite element formulations an estimate of the babuska brezzi stability constant an a priori lower bound estimate of the babuska brezzi bb inf sup discrete stability constant is given for general linear petrov galerkin formulations the estimate is based on reasonable assumptions regarding a specific auxiliary adjoint problem the similarities and discrepancies of the presented method compared to the well known fortin s criterion are also noted the estimate may be used to obtain numerical verification of the bb condition under the assumption that the finite element basis functions are almost orthogonal