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# FETI-DP preconditioners for a staggered discontinuous Galerkin formulation of the two-dimensional Stokes problem

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*in*Computers & Mathematics with Applications 68(12) · December 2014

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Abstract

In this paper, a class of FETI-DP preconditioners is developed for a fast solution of the linear system arising from staggered discontinuous Galerkin discretization of the two-dimensional Stokes equations. The discretization has been recently developed and has the distinctive advantages that it is optimally convergent and has a good local conservation property. In order to efficiently solve the linear system, two kinds of FETI-DP preconditioners, namely, lumped and Dirichlet preconditioners, are considered and analyzed. Scalable bounds C(H/h)C(H/h) and C(1+log(H/h))2C(1+log(H/h))2 are proved for the lumped and Dirichlet preconditioners, respectively, with the constant CC depending on the inf–sup constant of the discrete spaces but independent of any mesh parameters. Here H/hH/h stands for the number of elements across each subdomain. Numerical results are presented to confirm the theoretical estimates.

- ... An analysis of the SDG method for incompressible Navier-Stokes equations is given in [22]. For a more complete discussion on the SDG method, see also [13, 14, 15, 16, 20, 21, 32, 33] and the references therein. In the finite element formulation of IB method in [4], the convection term was neglected and linearized Navier-Stokes equations was considered. ...ArticleFull-text available
- Sep 2016

In this paper, we present a staggered discontinuous Galerkin immersed boundary method (SDG-IBM) for the numerical approximation of fluid-structure interaction. The immersed boundary method is used to model the fluid-structure interaction, while the fluid flow is governed by incompressible Navier-Stokes equations. One advantage of using Galerkin method over the finite difference method with immersed boundary method is that we can avoid approximations of the Dirac Delta function. Another key ingredient of our method is that our solver for incompressible Navier-Stokes equations combines the advantages of discontinuous Galerkin methods and staggered meshes, and results in many good properties, namely local and global conservations and pointwise divergence-free velocity field by a local postprocessing technique. Furthermore, energy stability is improved by a skew-symmetric discretization of the convection term. We will present numerical results to show the performance of the method. - ... Following [8,14151627,28], we first define the triangulation. For rectangular mesh, we can use the idea in [9]. ...
- ... Recently, a new class of discontinuous Galekin methods based on a novel type of staggered grid is introduced in Chung & Engquist [8, 9] for the wave equations, in Chung & Lee [12] and Chung & Kim [10] for the curl-curl operator, in Chung, Ciarlet & Yu [14] for Maxwell's equations, in Chung & Lee [13] for the convection-diffusion equation, in Kim, Chung & Lee [21] for the Stokes system and in Chan & Chung [5] for the Burgers equation. Moreover, wave transmission problems in the interface between classical material and meta-material using this kind of method is proposed and analyzed in Chung & Ciarlet [7], and fast solvers have been developed in Chung, Kim & Widlund [11], Kim, Chung & Lee [22] and Kim, Chung & Lee [23]. These methods have the advantages that the structures, such as energy and mass, arising from the partial differential equations are preserved. ...Article
- Sep 2015
- J COMPUT APPL MATH

In this paper, we develop and analyze a new class of spectral element methods for the simulations of elastic wave propagation. The major components of the method are the spatial discretization and the choice of interpolation nodes. The spatial discretization is based on piecewise polynomial approximation defined on staggered grids. The resulting method combines the advantages of both staggered-grid based methods and classical non-staggered-grid based spectral element methods. Our new method is energy conserving and does not require the use of any numerical flux, because of the staggered local continuity of the basis functions. Our new method also uses Radau points as interpolation nodes, and the resulting mass matrix is diagonal, thus time marching is explicit and is very efficient. Moreover, we give a rigorous proof for the optimal convergence of the method. In terms of dispersion, we present a numerical study for the numerical dispersion and show that this error is of very high order. Finally, some numerical convergence tests and applications to unbounded domain problems with perfectly matched layer are shown. - ... The above linear system is solved for them and then the unknowns for the velocity gradient are calculated from (6.1). We note that the linear system preserves the structure of the Stokes problem and existing domain decomposition algorithms for the Stokes problem can be easily applied for fast solutions, see [25]. The results for the numerical experiments are summarized in Tables 6.1-6.3. ...Discontinuous Galerkin (DG) methods are a cl ass of efficient tools for solving fluid flow problems. There are in the literature many greatly successful DG methods. In this paper, a new staggered DG method for the Stokes system is developed and analyzed. The key feature of our method is that the discrete system preserves the structures of the continuous problem, which results from the use of our new staggered DG spaces. This also provides local and global conservation properties, which are desirable for fluid flow applications. The method is based on the first order mixed formulation involving pressure, velocity, and velocity gradient. The velocity and velocity gradient are approximated by polynomials of the same degree while the choice of polynomial degree for pressure is flexible, namely, the approximation degree for pressure can be chosen as either that of velocity or one degree lower than that of velocity. In any case, stability and optimal convergence of the method are proved. Moreover, a superconvergence result with respect to a discrete H 1-norm for the velocity is proved. Furthermore, a local postprocessing technique is proposed to improve the divergence free property of the velocity approximation and it is proved that the postprocessed velocity retains the original accuracy and is weakly divergence free with respect to pressure test functions. Numerical results are included to validate our theoretical estimates and to present the ability of our method for capturing singular solutions.
- PreprintFull-text available
- Jun 2018

In this paper, we present an embedded staggered discontinuous Galerkin method for the convection-diffusion equation. The new method combines the advantages of staggered discontinuous Galerkin (SDG) and embedded discontinuous Galerkin (EDG) method, and results in many good properties, namely local and global conservations, free of carefully designed stabilization terms or flux conditions and high computational efficiency. In applying the new method to convection-dominated problems, the method provides optimal convergence in potential and suboptimal convergence in flux, which is comparable to other existing DG methods, and achieves $L^2$ stability by making use of a skew-symmetric discretization of the convection term, irrespective of diffusivity. We will present numerical results to show the performance of the method. - Article
- Jul 2017
- J SCI COMPUT

In this paper, we develop a new mass conservative numerical scheme for the simulations of a class of fluid–structure interaction problems. We will use the immersed boundary method to model the fluid–structure interaction, while the fluid flow is governed by the incompressible Navier–Stokes equations. The immersed boundary method is proven to be a successful scheme to model fluid–structure interactions. To ensure mass conservation, we will use the staggered discontinuous Galerkin method to discretize the incompressible Navier–Stokes equations. The staggered discontinuous Galerkin method is able to preserve the skew-symmetry of the convection term. In addition, by using a local postprocessing technique, the weakly divergence free velocity can be used to compute a new postprocessed velocity, which is exactly divergence free and has a superconvergence property. This strongly divergence free velocity field is the key to the mass conservation. Furthermore, energy stability is improved by the skew-symmetric discretization of the convection term. We will present several numerical results to show the performance of the method. - Article
- May 2016
- B KOREAN MATH SOC

Based on a particular overlapping domain decomposition technique, a parallel finite element discretization algorithm for the gener- alized Stokes equations is proposed and investigated. In this algorithm, each processor computes a local approximate solution in its own subdomain by solving a global problem on a mesh that is fine around its own subdomain and coarse elsewhere, and hence avoids communication with other processors in the process of computations. This algorithm has low communication complexity. It only requires the application of an existing sequential solver on the global meshes associated with each subdomain, and hence can reuse existing sequential software. Numerical results are given to demonstrate the effectiveness of the parallel algorithm.

- Article
- Apr 2012
- SIAM J NUMER ANAL

A non-overlapping domain decomposition algorithm is proposed to solve the linear system arising from mixed finite element approximation of incompressible Stokes equations. A continuous finite element space for the pressure is used. In the proposed algorithm, Lagrange multipliers are used to enforce continuity of the velocity component across the subdomain domain boundary. The continuity of the pressure component is enforced in the primal form, i.e., neighboring subdomains share the same pressure degrees of freedom on the subdomain interface and no Lagrange multipliers are needed. After eliminating all velocity variables and the independent subdomain interior parts of the pressures, a symmetric positive semi-definite linear system for the subdomain boundary pressures and the Lagrange multipliers is formed and solved by a preconditioned conjugate gradient method. A lumped preconditioner is studied and the condition number bound of the preconditioned operator is proved to be independent of the number of subdomains for fixed subdomain problem size. Numerical experiments demonstrate the convergence rate of the proposed algorithm. - Jan 2006
- 250-271

- J Li
- O B Widlund
- Feti-Dp
- Bddc
- Block Cholesky Methods
- Internat

J. Li, O.B. Widlund, FETI-DP, BDDC, and block Cholesky methods, Internat. J. Numer. Methods Engrg. 66 (2) (2006) 250–271.- Article
- Apr 2012
- J Numer Math

This paper is concerned with the staggered discontinuous Galerkin method for convection-diffusion equations. Over the past few decades, staggered type methods have been applied successfully to many problems, such as wave propagation and fluid flow problems. A distinctive feature of these methods is that the physical laws arising from the corresponding partial differential equations are automatically preserved. Nevertheless, staggered methods for convection-diffusion equations are rarely seen in literature. It is thus the main goal of this paper to develop and analyze a class of staggered numerical schemes for the approximation of convection-diffusion equations. We will prove that our new method preserves the underlying physical laws in some discrete sense.Moreover, the stability and convergence of the method are proved. Numerical results are shown to verify the theoretical estimates. - Article
- Jul 2012
- IMA J NUMER ANAL

This paper is concerned with a staggered discontinuous Galerkin method for the curl–curl operator arising from the time-harmonic Maxwell equations. One distinctive feature of the method is that the discrete operators preserve the properties of the differential operators. Moreover, the numerical solution automatically satisfies a discrete divergence-free condition. Stability and optimal convergence of the method are analysed. Numerical experiments for smooth and singular solutions are shown to verify the optimal order of convergence. Furthermore, the method is applied to the corresponding eigenvalue problem. Numerical results for rectangular and L-shaped domains show that our method is able to produce nonspurious eigenvalues. - Discontinuous Galerkin (DG) methods are a cl ass of efficient tools for solving fluid flow problems. There are in the literature many greatly successful DG methods. In this paper, a new staggered DG method for the Stokes system is developed and analyzed. The key feature of our method is that the discrete system preserves the structures of the continuous problem, which results from the use of our new staggered DG spaces. This also provides local and global conservation properties, which are desirable for fluid flow applications. The method is based on the first order mixed formulation involving pressure, velocity, and velocity gradient. The velocity and velocity gradient are approximated by polynomials of the same degree while the choice of polynomial degree for pressure is flexible, namely, the approximation degree for pressure can be chosen as either that of velocity or one degree lower than that of velocity. In any case, stability and optimal convergence of the method are proved. Moreover, a superconvergence result with respect to a discrete H 1-norm for the velocity is proved. Furthermore, a local postprocessing technique is proposed to improve the divergence free property of the velocity approximation and it is proved that the postprocessed velocity retains the original accuracy and is weakly divergence free with respect to pressure test functions. Numerical results are included to validate our theoretical estimates and to present the ability of our method for capturing singular solutions.
- Article
- Jan 2013
- INT J NUMER ANAL MOD

Staggered discontinuous Galerkin methods have been developed recently and are adopted successfully to many problems such as wave propagation, elliptic equations, convection-diffusion equations and the Maxwell’s equations. For wave propagation, the method is proved to have the desirable properties of energy conservation, optimal order of convergence and blockdiagonal mass matrices. In this paper, we perform an analysis for the dispersion error and the CFL constant. Our results show that the staggered method provides a smaller dispersion error compared with the classical finite element method as well as non-staggered discontinuous Galerkin methods. - Article
- Apr 2014
- COMPUT MATH APPL

A BDDC (Balancing Domain Decomposition by Constraints) algorithm is developed and analyzed for a staggered discontinuous Galerkin (DG) finite element approximation of second order scalar elliptic problems. On a quite irregular subdomain partition, an optimal condition number bound is proved for two-dimensional problems. In addition, a sub-optimal but scalable condition number bound is obtained for three-dimensional problems. These bounds are shown to be independent of coefficient jumps in the subdomain partition. Numerical results are also included to show the performance of the algorithm. - Article
- Apr 2013
- INT J NUMER METH ENG

SUMMARYA unified framework of dual‐primal finite element tearing and interconnecting (FETI‐DP) algorithms is proposed for solving the system of linear equations arising from the mixed finite element approximation of incompressible Stokes equations. A distinctive feature of this framework is that it allows using both continuous and discontinuous pressures in the algorithm, whereas previous FETI‐DP methods only apply to discontinuous pressures. A preconditioned conjugate gradient method is used in the algorithm with either a lumped or a Dirichlet preconditioner, and scalable convergence rates are proved. This framework is also used to describe several previously developed FETI‐DP algorithms and greatly simplifies their analysis. Numerical experiments of solving a two‐dimensional incompressible Stokes problem demonstrate the performances of the discussed FETI‐DP algorithms represented under the same framework.Copyright © 2012 John Wiley & Sons, Ltd. - Article
- Aug 2012
- J SCI COMPUT

In this paper we discuss the stability of some Stokes finite elements. In particular, we consider a modification of Hood–Taylor and Bercovier–Pironneau schemes which consists in adding piecewise constant functions to the pressure space. This enhancement, which had been already used in the literature, is driven by the goal of achieving an improved mass conservation at element level. The main result consists in proving the inf-sup condition for the enhanced spaces in a general setting and to present some numerical tests which confirm the stability properties. The improvement in the local mass conservation is shown in a forthcoming paper (Boffi et al. In: Papadrakakis, M., Onate, E., Schrefler, B. (eds.) Coupled Problems 2011. Computational Methods for Coupled Problems in Science and Engineering IV, Cimne, 2011) where the presented schemes are used for the solution of a fluid-structure interaction problem. - Article
- Feb 2013
- J COMPUT APPL MATH

Some electromagnetic materials exhibit, in a given frequency range, effective dielectric permittivity and/or magnetic permeability which are negative. In the literature, they are called negative index materials, left-handed materials or meta-materials. We propose in this paper a numerical method to solve a wave transmission between a classical dielectric material and a meta-material. The method we investigate can be considered as an alternative method compared to the method presented by the second author and co-workers. In particular, we shall use the abstract framework they developed to prove well-posedness of the exact problem. We recast this problem to fit later discretization by the staggered discontinuous Galerkin method developed by the first author and co-worker, a method which relies on introducing an auxiliary unknown. Convergence of the numerical method is proven, with the help of explicit inf–sup operators, and numerical examples are provided to show the efficiency of the method. - Article
- Feb 2013
- J COMPUT PHYS

In this paper, a new type of staggered discontinuous Galerkin methods for the three dimensional Maxwell’s equations is developed and analyzed. The spatial discretization is based on staggered Cartesian grids so that many good properties are obtained. First of all, our method has the advantages that the numerical solution preserves the electromagnetic energy and automatically fulfills a discrete version of the Gauss law. Moreover, the mass matrices are diagonal, thus time marching is explicit and is very efficient. Our method is high order accurate and the optimal order of convergence is rigorously proved. It is also very easy to implement due to its Cartesian structure and can be regarded as a generalization of the classical Yee’s scheme as well as the quadrilateral edge finite elements. Furthermore, a superconvergence result, that is the convergence rate is one order higher at interpolation nodes, is proved. Numerical results are shown to confirm our theoretical statements, and applications to problems in unbounded domains with the use of PML are presented. A comparison of our staggered method and non-staggered method is carried out and shows that our method has better accuracy and efficiency. - Article
- Apr 2002
- SIAM J NUMER ANAL

In this paper, we introduce and analyze local discontinuous Galerkin methods for the Stokes system. For arbitrary meshes with hanging nodes and elements of various shapes we derive a priori estimates for the L 2 -norm of the errors in the velocities and the pressure. We show that optimal order estimates are obtained when polynomials of degree k are used for each component of the velocity and polynomials of degree k 1 for the pressure, for any k 1. We also consider the case in which all the unknowns are approximated with polynomials of degree k and show that, although the orders of convergence remain the same, the method is more ecient. Numerical experiments verifying these facts are displayed. - Article
- Feb 2003
- SIAM J NUMER ANAL

Poincaré--Friedrichs inequalities for piecewise H1 functions are established. They can be applied to classical nonconforming finite element methods, mortar methods, and discontinuous Galerkin methods. - Article
- Mar 2002
- Comm Pure Appl Math

Balancing Neumann-Neumann methods are introduced and studied for incompressible Stokes equations discretized with mixed finite or spectral elements with discontinuous pressures. After decomposing the original domain of the problem into nonoverlapping subdomains, the interior unknowns, which are the interior velocity component and all except the constant-pressure component, of each subdomain problem are implicitly eliminated. The resulting saddle point Schur complement is solved with a Krylov space method with a balancing Neumann-Neumann preconditioner based on the solution of a coarse Stokes problem with a few degrees of freedom per subdomain and on the solution of local Stokes problems with natural and essential boundary conditions on the subdomains. This preconditioner is of hybrid form in which the coarse problem is treated multiplicatively while the local problems are treated additively. The condition number of the preconditioned operator is independent of the number of subdomains and is bounded from above by the product of the square of the logarithm of the local number of unknowns in each subdomain and a factor that depends on the inverse of the inf-sup constants of the discrete problem and of the coarse subproblem. Numerical results show that the method is quite fast; they are also fully consistent with the theory. © 2002 John Wiley & Sons, Inc. - Article
- Sep 1989
- NUMER MATH

We present the convergence analysis of a new domain decomposition technique for finite element approximations. This technique was introduced in [11] and is based on an iterative procedure among subdomains in which transmission conditions at interfaces are taken into account partly in one subdomain and partly in its adjacent. No global preconditioner is needed in the practice, but simply single-domain finite element solvers are required. An optimal strategy for an automatic selection of a relaxation parameter to be used at interface subdomains is indicated. Applications are given to both elliptic equations and incompressible Stokes equations. - Article
- Jan 2000
- Numer Lin Algebra Appl

Three domain decomposition methods for saddle point problems are introduced and compared. The first two are block-diagonal and block-triangular preconditioners with diagonal blocks approximated by an overlapping Schwarz technique with positive definite local and coarse problems. The third is an overlapping Schwarz preconditioner based on indefinite local and coarse problems. Numerical experiments show that while all three methods are numerically scalable, the last method is almost always the most efficient. - Balancing Neumann-Neumann methods are extented to mixed formulations of the linear elasticity system with discontinuous coefficients, discretized with mixed finite or spectral elements with discontinuous pressures. These domain decomposition methods implicitly eliminate the degrees of freedom associated with the interior of each subdomain and solve iteratively the resulting saddle point Schur complement using a hybrid preconditioner based on a coarse mixed elasticity problem and local mixed elasticity problems with natural and essential boundary conditions. A polylogarithmic bound in the local number of degrees of freedom is proven for the condition number of the preconditioned operator in the constant coefficient case. Parallel and serial numerical experiments confirm the theoretical results, indicate that they still hold for systems with discontinuous coefficients, and show that our algorithm is scalable, parallel, and robust with respect to material heterogeneities. The results on heterogeneous general problems are also supported in part by our theory.
- Article
- Dec 2010
- COMPUT MATH APPL

Selection of primal unknowns is important in convergence of FETI-DP (dual-primal finite element tearing and interconnecting) methods, which are known to be the most scalable dual iterative substructuring methods. A FETI-DP algorithm for the Stokes problem without primal pressure unknowns was developed and analyzed by Kim et al. (2010) [1]. Only the velocity unknowns at the subdomain vertices are selected to be the primal unknowns and convergence of the algorithm with a lumped preconditioner is determined by the condition number bound C(H/h)(1+log(H/h)), where H/h is the number of elements across subdomains. In this work, primal unknowns corresponding to the averages on edges are introduced and a better condition number bound C(H/h) is proved for such a selection of primal unknowns. Numerical results are included. - Article
- May 1990
- APPL NUMER MATH

In this paper, we give an analysis for a domain decomposition technique for Stokes problems. The technique involves the application of domain decomposition directly to the Stokes problem and gives rise to an indifinite system for the velocity nodes on the subdomain boundaries and the mean values of the pressure on the subdomains. We analyze the resulting system and show how it can be efficiently solved. - Article
- Jul 2011
- COMPUT FLUIDS

A parallel implementation of the Balancing Domain Decomposition by Constraints (BDDC) method is described. It is based on formulation of BDDC with global matrices without explicit coarse problem. The implementation is based on the MUMPS parallel solver for computing the approximate inverse used for preconditioning. It is successfully applied to several problems of Stokes flow discretized by Taylor–Hood finite elements and BDDC is shown to be a promising method also for this class of problems. - Article
- Nov 1998
- COMPUT METHOD APPL M

Overlapping Schwarz preconditioners are introduced and studied for saddle point problems with a penalty term, such as Stokes equations and mixed formulations of linear elasticity. These preconditioners are based on the solution of local saddle point problems on overlapping subdomains and the solution of a coarse saddle point problem. Numerical experiments show that these are parallel and scalable preconditioners, since the rate of convergence of the preconditioned operator is independent of the mesh size h, the number of subdomains N, and the penalty parameter. - Article
- Jan 2007
- COMPUT METHOD APPL M

The standard BDDC (balancing domain decomposition by constraints) preconditioner is shown to be equivalent to a preconditioner built from a partially subassembled finite element model. This results in a system of linear algebraic equations which is much easier to solve in parallel than the fully assembled model; the cost is then often dominated by that of the problems on the subdomains. An important role is also played, both in theory and practice, by an averaging operator and in addition exact Dirichlet solvers are used on the subdomains in order to eliminate the residual in the interior of the subdomains. The use of inexact solvers for these problems and even the replacement of the Dirichlet solvers by a trivial extension are considered. It is established that one of the resulting algorithms has the same eigenvalues as the standard BDDC algorithm, and the connection of another with the FETI-DP algorithm with a lumped preconditioner is also considered. Multigrid methods are used in the experimental work and under certain assumptions, it is established that the iteration count essentially remains the same as when exact solvers are used, while considerable gains in the speed of the algorithm can be realized since the cost of the exact solvers grows superlinearly with the size of the subdomain problems while the multigrid methods are linear. - Article
- Jun 1997
- COMPUT METHOD APPL M

This paper proposes a framework for non-overlapping domain decomposition methods applied to viscous incompressible flows governed by Stokes equations modelled by the hp version of the finite element method. Preconditioning methods developed earlier for elliptic problems are extended to this case. Suitable condition number bounds are obtained for both the hierarchical hp preconditioners of the type proposed in [9] and Neumann-Neumann preconditioners proposed in [10]. Systematic procedures for constructing compatible hp approximations for incompressible flow are also described. - Article
- Apr 2006
- MATH COMPUT

We devise and analyze a new local discontinuous Galerkin (LDG) method for the Stokes equations of incompressible fluid flow. This optimally convergent method is obtained by using an LDG method to discretize a vorticity-velocity formulation of Stokes equations and by applying a new hybridization to the resulting discretization. One of the main features of the hybridized method is that it provides a globally divergence-free approximate velocity without having to construct globally divergence-free finite-dimensional spaces; only elementwise divergence-free basis functions are used. Another important feature is that it has significantly less degrees of freedom than all other LDG methods in the current literature; in particular, the approximation to the pressure is only defined on the faces of the elements. On the other hand, we show that, as expected, the condition number of the Schur-complement matrix for this approximate pressure is of order h -2 in the mesh size h. Finally, we present numerical experiments that confirm the sharpness of our theoretical a priori error estimates. - ArticleFull-text available
- Apr 2011

Seismic data are routinely used to infer in situ properties of earth materials on many scales, ranging from global studies to investigations of surficial geological formations. While inversion and imaging algorithms utilizing these data have improved steadily, there are remaining challenges that make detailed measurements of the properties of some geologic materials very difficult. For example, the determination of the concentration and orientation of fracture systems is prohibitively expensive to simulate on the fine grid and, thus, some type of coarse-grid simulations are needed. In this paper, we describe a new multiscale finite element algorithm for simulating seismic wave propagation in heterogeneous media. This method solves the wave equation on a coarse grid using multiscale basis functions and a global coupling mechanism to relate information between fine and coarse grids. Using a mixed formulation of the wave equation and staggered discontinuous basis functions, the proposed multiscale methods have the following properties. • The total wave energy is conserved. • Mass matrix is diagonal on a coarse grid and explicit energy-preserving time discretization does not require solving a linear system at each time step. • Multiscale basis functions can accurately capture the subgrid variations of the solution and the time stepping is performed on a coarse grid. We discuss various subgrid capturing mechanisms and present some preliminary numerical results. - Article
- Jan 2010
- SIAM J SCI COMPUT

A scalable FETI-DP (Dual-Primal Finite Element Tearing and Interconnecting) algorithm for the three-dimensional Stokes problem is developed and analyzed. This is an extension of the previous work for the two- dimensional problem in (8). Advantages of this approach are the coarse problem without primal pressure unknowns and the use of a relatively cheap lumped preconditioner. Especially in three dimensions, these advantages provide a more robust and faster FETI-DP algorithm. In three dimensions, the velocity unknowns at subdomain corners and the averages of velocity unknowns over common faces are selected as the primal unknowns in the FETI-DP formulation. Its condition number bound is analyzed to be C(H=h), where C is a positive constant which is independent of any mesh parameters and H=h is the number of elements across each subdomain. Numerical results are included. - Article
- Jan 2006
- SIAM J SCI COMPUT

A FETI-DP (dual-primal finite element tearing and interconnecting) formulation for the two-dimensional Stokes problem with mortar methods is considered. Separate sets of unknowns are used for velocity on interfaces, and the mortar constraints are enforced on the velocity unknowns by Lagrange multipliers. Average constraints on edges are further introduced as primal constraints to solve the Stokes problem correctly and to obtain a scalable FETI-DP algorithm. A Neumann-Dirichlet preconditioner is shown to give a condition number bound, C maxi=1,...,N{(1 + log (Hi/h i))2}, where Hi and hi are the subdomain size and the mesh size, respectively, and the constant C is independent of the mesh parameters Hi and hi. - Article
- Jun 2005
- J SCI COMPUT

We present some two-level non-overlapping additive and multiplicative Schwarz methods for a discontinuous Galerkin method for solving the biharmonic equation. We show that the condition numbers of the preconditioned systems are of the order O( H3/h3) for the non-overlapping Schwarz methods, where h and H stand for the fine mesh size and the coarse mesh size, respectively. The analysis requires establishing an interpolation result for Sobolev norms and Poincaré–Friedrichs type inequalities for totally discontinuous piecewise polynomial functions. It also requires showing some approximation properties of the multilevel hierarchy of discontinuous Galerkin finite element spaces. - In this paper, we develop and analyze a new class of discontinuous Galerkin (DG) methods for the acoustic wave equation in mixed form. Traditional mixed finite element (FE) methods produce energy conserving schemes, but these schemes are implicit, making the time-stepping inefficient. Standard DG methods give explicit schemes, but these approaches are typically dissipative or suboptimally convergent, depending on the choice of numerical fluxes. Our new method can be seen as a compromise between these two kinds of techniques, in the way that it is both explicit and energy conserving, locally and globally. Moreover, it can be seen as a generalized version of the Raviart-Thomas FE method and the finite volume method. Stability and convergence of the new method are rigorously analyzed, and we have shown that the method is optimally convergent. Furthermore, in order to apply the new method for unbounded domains, we apply our new method with the first order absorbing boundary condition. The stability of the resulting numerical scheme is analyzed.
- Article
- Jan 2009
- SIAM J NUMER ANAL

Overlapping Schwarz methods are extended to mixed finite element approximations of linear elasticity which use discontinuous pressure spaces. The coarse component of the preconditioner is based on a low-dimensional space previously developed for scalar elliptic problems and a domain decomposition method of iterative substructuring type, i.e., a method based on nonoverlapping decompositions of the domain, while the local components of the preconditioner are based on solvers on a set of overlapping subdomains. A bound is established for the condition number of the algorithm which grows in proportion to the logarithm of the number of degrees of freedom in individual subdomains and, essentially, to the third power of the relative overlap between the overlapping subdomains, and which is independent of the Poisson ratio as well as jumps in the Lamé parameters across the interface between the subdomains. A positive definite reformulation of the discrete problem makes the use of the standard preconditioned conjugate gradient method straightforward. Numerical results, which include a comparison with problems of compressible elasticity, illustrate the findings. - Article
- Jan 2010
- SIAM J NUMER ANAL

A scalable FETI–DP (Dual-Primal Finite Element Tearing and Interconnecting) algorithm for the Stokes problem is developed and analyzed. Advantages of this approach are a coarse problem without primal pressure unknowns and the use of a relatively cheap lumped preconditioner. Especially in three dimensions, these advantages provide a more robust and faster FETI-DP algorithm. In three dimensions, the velocity unknowns at subdomain corners and the averages of velocity unknowns over common faces are selected as the primal unknowns in the FETI-DP formulation. A condition number bound of the form C (H/h) is established, where C is a positive constant which is independent of any mesh parameters and H/h is the number of elements across individual subdomains. - Article
- May 2011
- MATH COMPUT

In this paper, we analyze a hybridizable discontinuous Galerkin method for numerically solving the Stokes equations. The method uses polynomials of degree k for all the components of the approximate solution of the gradient-velocity-pressure formulation. The novelty of the analysis is the use of a new projection tailored to the very structure of the numerical traces of the method. It renders the analysis of the projection of the errors very concise and allows us to see that the projection of the error in the velocity superconverges. As a consequence, we prove that the approximations of the velocity gradient, the velocity and the pressure converge with the optimal order of convergence of k+1 in L(2) for any k >= 0. Moreover, taking advantage of the superconvergence properties of the velocity, we introduce a new element-by-element postprocessing to obtain a new velocity approximation which is exactly divergence-free, H(div)-conforming, and converges with order k + 2 for k >= 1 and with order 1 for k = 0. Numerical experiments are presented which validate the theoretical results. - In this paper a new local discontinuous Galerkin method for the incompressible stationary Navier-Stokes equations is proposed and analyzed. Four important features render this method unique: its stability, its local conservativity, its high-order accuracy, and the exact satisfaction of the incompressibility constraint. Although the method uses completely discontinuous approximations, a globally divergence-free approximate velocity in H(div; Omega) is obtained by simple, element-by-element post-processing. Optimal error estimates are proven and an iterative procedure used to compute the approximate solution is shown to converge. This procedure is nothing but a discrete version of the classical fixed point iteration used to obtain existence and uniqueness of solutions to the incompressible Navier-Stokes equations by solving a sequence of Oseen problems. Numerical results are shown which verify the theoretical rates of convergence. They also confirm the independence of the number of fixed point iterations with respect to the discretization parameters. Finally, they show that the method works well for a wide range of Reynolds numbers.
- Article
- Apr 2006
- INT J NUMER METH ENG

The FETI-DP and BDDC algorithms are reformulated using Block Cholesky factorizations, an approach which can provide a useful framework for the design of domain decomposition algorithms for solving symmetric positive definite linear system of equations. Instead of introducing Lagrange multipliers to enforce the coarse level, primal continuity constraints in these algorithms, a change of variables is used such that each primal constraint corresponds to an explicit degree of freedom. With the new formulation of these algorithms, a simplified proof is provided that the spectra of a pair of FETI-DP and BDDC algorithms, with the same set of primal constraints, are essentially the same. Numerical experiments for a two-dimensional Laplace's equation also confirm this result. - Article
- Jan 2006
- SIAM J NUMER ANAL

The purpose of this paper is to extend the BDDC (balancing domain decomposition by constraints) algorithm to saddle-point problems that arise when mixed finite element methods are used to approximate the system of incompressible Stokes equations. The BDDC algorithms are iterative substructuring methods, which form a class of domain decomposition methods based on the decomposition of the domain of the dierential equations into nonoverlapping subdomains. They are defined in terms of a set of primal continuity constraints, which are enforced across the interface between the subdomains and which provide a coarse space component of the preconditioner. Sets of such constraints are identified for which bounds on the rate of convergence can be established that are just as strong as previously known bounds for the elliptic case. In fact, the preconditioned operator is eectively positive definite, which makes the use of a conjugate gradient method possible. A close connection is also established between the BDDC and FETI-DP algorithms for the Stokes case. - We have developed and analyzed a new class of discontinuous Galerkin methods (DG) which can be seen as a compromise between standard DG and the finite element (FE) method in the way that it is explicit like standard DG and energy conserving like FE. In the literature there are many methods that achieve some of the goals of explicit time marching, unstructured grid, energy conservation, and optimal higher order accuracy, but as far as we know only our new algorithms satisfy all the conditions. We propose a new stability requirement for our DG. The stability analysis is based on the careful selection of the two FE spaces which verify the new stability condition. The convergence rate is optimal with respect to the order of the polynomials in the FE spaces. Moreover, the convergence is described by a series of numerical experiments.
- Article
- Feb 2004

Balancing Neumann-Neumann methods are extended to the equations arising from the mixed formulation of almost-incompressible linear elasticity problems discretized with discontinuous-pressure finite elements. This family of domain decomposition algorithms has previously been shown to be e#ective for large finite element approximations of positive definite elliptic problems. Our methods are proved to be scalable and to depend weakly on the size of the local problems. Our work is an extension of previous work by Pavarino and Widlund on BNN methods for Stokes equation. - Article
- Sep 2001
- NUMER MATH

. In this paper, a dual-primal FETI method is developed for incompressible Stokes equations approximated by mixed nite elements with discontinuous pressures. The domain of the problem is decomposed into nonoverlapping subdomains, and the continuity of the velocity across the subdomain interface is enforced by introducing Lagrange multipliers. By a Schur complement procedure, solving the indenite Stokes problem is reduced to solving a symmetric positive denite problem for the dual variables, i.e., the Lagrange multipliers. This dual problem is solved by a Krylov space method with a Dirichlet preconditioner. At each step of the iteration, both subdomain problems and a coarse problem on the coarse subdomain mesh are solved by a direct method. It is proved that the condition number of this preconditioned dual problem is independent of the number of subdomains and bounded from above by the product of the inverse of the inf-sup constant of the discrete problem and the square of the logarithm of the number of unknowns in the individual subdomain problems. Illustrative numerical results are presented by solving a lid driven cavity problem. Key words. domain decomposition, Stokes, FETI, dual-primal methods AMS subject classications. 65N30, 65N55, 76D07 1.