Double Greedy Algorithms: Reduced Basis Methods for Transport Dominated Problems

The central objective of this paper is to develop reduced basis methods for parameter dependent transport dominated problems that are rigorously proven to exhibit rate-optimal performance when compared with the Kolmogorov $n$-widths of the solution sets. The central ingredient is the construction of computationally feasible "tight" surrogates which in turn are based on deriving a suitable well-conditioned variational formulation for the parameter dependent problem. The theoretical results are illustrated by numerical experiments for convection-diffusion and pure transport equations. In particular, the latter example sheds some light on the smoothness of the dependence of the solutions on the parameters

Hybrid First-Order System Least-Squares Finite Element Methods With The Application To Stokes And Navier-Stokes Equations

This thesis combines the FOSLS method with the FOSLL* method to create a Hybrid method. The FOSLS approach minimizes the error, e h = uh − u, over a finite element subspace, [special characters omitted], in the operator norm, [special characters omitted] ||L(uh − u)||. The FOSLL* method looks for an approximation in the range of L*, setting uh = L*wh and choosing wh ∈ [special characters omitted], a standard finite element space. FOSLL* minimizes the L 2 norm of the error over L*([special characters omitted]), that is, [special characters omitted] ||L*wh − u||. FOSLS enjoys a locally sharp, globally reliable, and easily computable a posterior error estimate, while FOSLL* does not. The Hybrid method attempts to retain the best properties of both FOSLS and FOSLL*. This is accomplished by combining the FOSLS functional, the FOSLL* functional, and an intermediate term that draws them together. The Hybrid method produces an approximation, uh, that is nearly the optimal over [special characters omitted] in the graph norm, ||eh[special characters omitted] := ½||eh|| 2 + ||Leh|| 2. The FOSLS and intermediate terms in the Hybrid functional provide a very effective a posteriori error measure. In this dissertation we show that the Hybrid functional is coercive and continuous in graph-like norm with modest coercivity and continuity constants, c0 = 1/3 and c1 = 3; that both ||eh|| and ||L eh|| converge with rates based on standard interpolation bounds; and that, if LL* has full H2-regularity, the L2 error, ||eh||, converges with a full power of the discretization parameter, h, faster than the functional norm. Letting ũh denote the optimum over [special characters omitted] in the graph norm, we also show that if superposition is used, then ||uh − ũ h[special characters omitted] converges two powers of h faster than the functional norm. Numerical tests on are provided to confirm the efficiency of the Hybrid method and effectiveness of the a posteriori error measure

Dual-Norm Least-Squares Finite Element Methods for Hyperbolic Problems

Least-squares finite element discretizations of first-order hyperbolic partial differential equations (PDEs) are proposed and studied.Hyperbolic problems are notorious for possessing solutions with jump discontinuities, like contact discontinuities and shocks, and steep exponential layers. Furthermore, nonlinear equations can have rarefaction waves as solutions. All these contribute to the challenges in the numerical treatment of hyperbolic PDEs.&nbsp; The approach here is to obtain appropriate least-squares formulations based on suitable minimization principles. Typically, such formulations can be reduced to one or more (e.g., by employing a Newton-type linearization procedure) quadratic minimization problems. Both theory and numerical results are presented. A method for nonlinear hyperbolic balance and conservation laws is proposed. The formulation is based on a Helmholtz decomposition and closely related to the notion of a weak solution and a H⁻&sup1;-type least-squares principle. Accordingly, the respective important conservation properties are studied in detail and the theoretically challenging convergence properties, with respect to the L&sup2; norm, are discussed. In the linear case, the convergence in the L&sup2; norm is explicitly and naturally guaranteed by suitable formulations that are founded upon the original LL* method developed for elliptic PDEs. The approaches considered here are the LL*-based and LL*⁻&sup1; methods, where the latter utilizes a special negative-norm least-squares minimization principle. These methods can be viewed as specific approximations of the generally infeasible quadratic minimization that determines the L&sup2;-orthogonal projection of the exact solution. The formulations are analyzed and studied in detail. &nbsp;</p

Dual-Norm Least-Squares Finite Element Methods for Hyperbolic Problems

Least-squares finite element discretizations of first-order hyperbolic partial differential equations (PDEs) are proposed and studied. Hyperbolic problems are notorious for possessing solutions with jump discontinuities, like contact discontinuities and shocks, and steep exponential layers. Furthermore, nonlinear equations can have rarefaction waves as solutions. All these contribute to the challenges in the numerical treatment of hyperbolic PDEs. The approach here is to obtain appropriate least-squares formulations based on suitable minimization principles. Typically, such formulations can be reduced to one or more (e.g., by employing a Newton-type linearization procedure) quadratic minimization problems. Both theory and numerical results are presented. A method for nonlinear hyperbolic balance and conservation laws is proposed. The formulation is based on a Helmholtz decomposition and closely related to the notion of a weak solution and a $H^{-1}$-type least-squares principle. Accordingly, the respective important conservation properties are studied in detail and the theoretically challenging convergence properties, with respect to the $L^2$ norm, are discussed. In the linear case, the convergence in the $L^2$ norm is explicitly and naturally guaranteed by suitable formulations that are founded upon the original \cL\cL^\star method developed for elliptic PDEs. The approaches considered here are the \cL\cL^\star-based and (\cL\cL^\star)^{-1} methods, where the latter utilizes a special negative-norm least-squares minimization principle. These methods can be viewed as specific approximations of the generally infeasible quadratic minimization that determines the $L^2$-orthogonal projection of the exact solution. The formulations are analyzed and studied in detail. Key words: first-order hyperbolic problems; hyperbolic balance laws; Burgers equation; weak solutions; Helmholtz decomposition; conservation properties; space-time discretization; least-squares methods; dual methods; adjoint methods; negative-norm methods; finite element methods; discontinuous coefficients; exponential layers AMS subject classifications: 65N30, 65N1