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38
Parallel Multilevel Methods With Adaptivity on Unstructured Grids
, 1998
"... Two parallel multilevel methods are presented for solving large discretized partial differential equations on unstructured 2D/3D grids. In short, the presented methods combine three powerful numerical algorithms, i.e., overlapping domain decomposition, multigrid method and adaptivity. As the foundat ..."
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Two parallel multilevel methods are presented for solving large discretized partial differential equations on unstructured 2D/3D grids. In short, the presented methods combine three powerful numerical algorithms, i.e., overlapping domain decomposition, multigrid method and adaptivity. As the foundation of the methods we propose an algorithm for generating and partitioning a hierarchy of adaptively refined unstructured grids, so that adaptivity can be incorporated up to a certain grid level and the resulting subgrids are wellbalanced. The first method uses a domain decomposition approach where we replace the singlelevel coarse grid in standard two level overlapping Schwarz methods by a hierarchy of adaptively refined unstructured grids. Efficient coarse grid correction is done by multigrid Vcycles. Moreover, multigrid Vcycles are also used in local subproblem solves. The second method uses a multigrid approach where we parallelize global multigrid Vcycles on all the grid levels. Nu...
Adaptive finite element methods for mixed controlstate constrained optimal control problems for elliptic boundary value problems
 Computational Optimization and Applications
"... Abstract. Mixed controlstate constraints are used as a relaxation of originally state constrained optimal control problems for partial differential equations to avoid the intrinsic difficulties arising from measurevalued multipliers in the case of pure state constraints. In particular, numerical s ..."
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Abstract. Mixed controlstate constraints are used as a relaxation of originally state constrained optimal control problems for partial differential equations to avoid the intrinsic difficulties arising from measurevalued multipliers in the case of pure state constraints. In particular, numerical solution techniques known from the pure control constrained case such as active set strategies and interiorpoint methods can be used in an appropriately modified way. However, the residualtype a posteriori error estimators developed for the pure control constrained case can not be applied directly. It is the essence of this paper to show that instead one has to resort to that type of estimators known from the pure state constrained case. Up to data oscillations and consistency error terms, they provide efficient and reliable estimates for the discretization errors in the state, a regularized adjoint state, and the control. A documentation of numerical results is given to illustrate the performance of the estimators.
Leastsquares finite element methods and algebraic multigrid solvers for linear hyperbolic PDEs
 SIAM J. Sci. Comput
"... Abstract. Leastsquares finite element methods (LSFEMs) for scalar linear partial differential equations (PDEs) of hyperbolic type are studied. The space of admissible boundary data is identified precisely, and a trace theorem and a Poincaré inequality are formulated. The PDE is restated as the mini ..."
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Abstract. Leastsquares finite element methods (LSFEMs) for scalar linear partial differential equations (PDEs) of hyperbolic type are studied. The space of admissible boundary data is identified precisely, and a trace theorem and a Poincaré inequality are formulated. The PDE is restated as the minimization of a leastsquares functional, and the wellposedness of the associated weak formulation is proved. Finite element convergence is proved for conforming and nonconforming (discontinuous) LSFEMs that are similar to previously proposed methods but for which no rigorous convergence proofs have been given in the literature. Convergence properties and solution quality for discontinuous solutions are investigated in detail for finite elements of increasing polynomial degree on triangular and quadrilateral meshes and for the general case that the discontinuity is not aligned with the computational mesh. Our numerical studies found that higherorder elements yield slightly better convergence properties when measured in terms of the number of degrees of freedom. Standard algebraic multigrid methods that are known to be optimal for large classes of elliptic PDEs are applied without modifications to the linear systems that result from the hyperbolic LSFEM formulations. They are found to yield complexity that grows only slowly relative to the size of the linear systems. Key words. leastsquares variational formulation, finite element discretization, hyperbolic problems, algebraic multigrid
CONSTRAINED DIRICHLET BOUNDARY CONTROL IN L 2 FOR A CLASS OF EVOLUTION EQUATIONS
"... Abstract. Optimal Dirichlet boundary control based on the very weak solution of a parabolic state equation is analysed. This approach allows to consider the boundary controls in L2 which has advantages over approaches which consider control in Sobolev involving (fractional) derivatives. Pointwise c ..."
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Abstract. Optimal Dirichlet boundary control based on the very weak solution of a parabolic state equation is analysed. This approach allows to consider the boundary controls in L2 which has advantages over approaches which consider control in Sobolev involving (fractional) derivatives. Pointwise constraints on the boundary are incorporated by the primaldual active set strategy. Its global and local superlinear convergence are shown. A discretization based on spacetime finite elements is proposed and numerical examples are included. Key words. Dirichlet boundary control, inequality constraints, parabolic equations, very weak solution 1. Introduction. In
A Comparison of Hereditary Integral and Internal Variable Approaches to Numerical Linear Solid Viscoelasticity
, 1997
"... This article first compares the mathematical models obtained when a problem of linear quasistatic solid viscoelasticity is modelled by: (i) a hereditary integral with Prony series; and, (ii) an evolution equation for internal strain variables. We then confirm that while the models are the same, the ..."
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This article first compares the mathematical models obtained when a problem of linear quasistatic solid viscoelasticity is modelled by: (i) a hereditary integral with Prony series; and, (ii) an evolution equation for internal strain variables. We then confirm that while the models are the same, the numerical approximations arising from them will, in general, be different. This suggests the need for a posteriori error estimates for the internal variable formulation so that viscoelasticity problems may be efficiently and reliably simulated. Acknowledgement The authors are pleased to acknowledge the stimulation of their interest in internal variable methods in the viscoelasticity context derived from conversations with Dr A R Johnson. 1 Introduction
Using KrylovSubspace Iterations in Discontinuous Galerkin Methods for Nonlinear ReactionDiffusion Systems
, 1999
"... We consider discontinuous in time and continuous in space Galerkin finiteelement methods for the numerical solution of reactiondiffusion differential equations. These are implicit methods that require the solution of a system of nonlinear equations at each time node. In this paper, we explore the ..."
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We consider discontinuous in time and continuous in space Galerkin finiteelement methods for the numerical solution of reactiondiffusion differential equations. These are implicit methods that require the solution of a system of nonlinear equations at each time node. In this paper, we explore the use of Krylovsubspace techniques for the iterative solution of the linear systems that arise when these nonlinear systems are solved by means of Newtontype methods. It is shown how these linear systems depend on the choice of the basis functions used for the time discretization. We demonstrate that Krylovsubspace methods can be sped up considerably by employing an orthogonal basis for the time discretization and by combining the Krylov iteration with a suitable block preconditioner. Results of numerical experiments are reported.
From Functional Analysis to Iterative Methods
 SIAM Review
"... Abstract. We examine condition numbers, preconditioners, and iterative methods for finite element discretizations of coercive PDEs in the context of the fundamental solvability result, the LaxMilgram Lemma. Working in this Hilbert space context is justified because finite element operators are rest ..."
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Abstract. We examine condition numbers, preconditioners, and iterative methods for finite element discretizations of coercive PDEs in the context of the fundamental solvability result, the LaxMilgram Lemma. Working in this Hilbert space context is justified because finite element operators are restrictions of infinitedimensional Hilbert space operators to finitedimensional subspaces. Moreover, useful insight is gained as to the relationship between Hilbert space and matrix condition numbers, and translating Hilbert space fixed point iterations into matrix computations provides new ways of motivating and explaining some classic iteration schemes. In this framework, the “simplest” preconditioner for an operator from a Hilbert space into its dual is the Riesz isomorphism. Simple analysis gives spectral bounds and iteration counts bounded independent of the finite element subspaces chosen. Moreover, the abstraction allows us not only to consider Riesz map preconditioning for convectiondiffusion equations in H1, but also operators on other Hilbert spaces, such as planar elasticity in ` H1 ´ 2
MODELING OF ATMOSPHERIC TRANSPORT OF CHEMICAL SPECIES IN THE POLAR REGIONS
"... This work is concerned with the reconstruction of unknown atmospheric distributions of species ’ concentrations and the simultaneous identification of unknown physical parameters. The physical and chemical processes are represented by a simplified chemical transport model. The second focus of the co ..."
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This work is concerned with the reconstruction of unknown atmospheric distributions of species ’ concentrations and the simultaneous identification of unknown physical parameters. The physical and chemical processes are represented by a simplified chemical transport model. The second focus of the contribution lies on the development and the application of the adaptive techniques to the resulting inverse problem. The underlying theoretical framework is the Dual Weighted Residual (DWR) method for goaloriented mesh optimization. Index Terms — Atmospheric modeling, inverse problems, goaloriented adaptivity, mesh optimization.
Topics: Numerical solution of elliptic, parabolic, and hyperbolic partial
"... Email is welcome anytime! Prerequisite: Undergraduatelevel knowledge of numerical analysis. Programming assignments will be in Matlab and will use the PDE Toolpack. ..."
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Email is welcome anytime! Prerequisite: Undergraduatelevel knowledge of numerical analysis. Programming assignments will be in Matlab and will use the PDE Toolpack.
The Residual–Free Bubble Method
"... This thesis is devoted to the numerical analysis and development of the residual–free bubble finite element method. We begin with an overview of known results and properties of the method, showing how techniques used on a range of multiscale problems can be cast into the framework of the residual–fr ..."
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This thesis is devoted to the numerical analysis and development of the residual–free bubble finite element method. We begin with an overview of known results and properties of the method, showing how techniques used on a range of multiscale problems can be cast into the framework of the residual–free bubble method. Further, we present an a priori error analysis of the method applied to convection–dominated diffusion problems on anisotropic meshes. The result has implications for the problem of parameter–tuning in classical stabilised finite element methods (for instance, the streamline– diffusion finite element method). We show how the local SD–parameter should be chosen on meshes with high aspect–ratios. A new algorithm named RFBe (enhanced residual–free bubble method) is proposed for the resolution of boundary layers on coarse meshes. The residual–free bubble finite element space is augmented locally by ad hoc bubble functions with support on two elements sharing a particular edge. The idea is presented in a general framework to highlight its applicability to a wide range of multiscale problems.