12 research outputs found
Quasi-optimal nonconforming methods for symmetric elliptic problems. I -- Abstract theory
We consider nonconforming methods for symmetric elliptic problems and
characterize their quasi-optimality in terms of suitable notions of stability
and consistency. The quasi-optimality constant is determined and the possible
impact of nonconformity on its size is quantified by means of two alternative
consistency measures. Identifying the structure of quasi-optimal methods, we
show that their construction reduces to the choice of suitable linear operators
mapping discrete functions to conforming ones. Such smoothing operators are
devised in the forthcoming parts of this work for various finite element
spaces
Time-parallel iterative solvers for parabolic evolution equations
We present original time-parallel algorithms for the solution of the implicit
Euler discretization of general linear parabolic evolution equations with
time-dependent self-adjoint spatial operators. Motivated by the inf-sup theory
of parabolic problems, we show that the standard nonsymmetric time-global
system can be equivalently reformulated as an original symmetric saddle-point
system that remains inf-sup stable with respect to the same natural parabolic
norms. We then propose and analyse an efficient and readily implementable
parallel-in-time preconditioner to be used with an inexact Uzawa method. The
proposed preconditioner is non-intrusive and easy to implement in practice, and
also features the key theoretical advantages of robust spectral bounds, leading
to convergence rates that are independent of the number of time-steps, final
time, or spatial mesh sizes, and also a theoretical parallel complexity that
grows only logarithmically with respect to the number of time-steps. Numerical
experiments with large-scale parallel computations show the effectiveness of
the method, along with its good weak and strong scaling properties
Quasi-optimal nonconforming methods for symmetric elliptic problems. I—abstract theory
We consider nonconforming methods for symmetric elliptic problems and characterize their quasi-optimality in terms of suitable notions of stability and consistency. The quasi-optimality constant is determined, and the possible impact of nonconformity on its size is quantified by means of two alternative consistency measures. Identifying the structure of quasi-optimal methods, we show that their construction reduces to the choice of suitable linear operators mapping discrete functions to conforming ones. Such smoothing operators are devised in the forthcoming parts of this work for various finite element spaces
Quasi-best approximation in optimization with PDE constraints
We consider finite element solutions to quadratic optimization problems, where the state depends on the control via a well-posed linear partial differential equation. Exploiting the structure of a suitably reduced optimality system, we prove that the combined error in the state and adjoint state of the variational discretization is bounded by the best approximation error in the underlying discrete spaces. The constant in this bound depends on the inverse square-root of the Tikhonov regularization parameter. Furthermore, if the operators of control-action and observation are compact, this quasibest-approximation constant becomes independent of the Tikhonov parameter as the meshsize tends to 0 and we give quantitative relationships between meshsize and Tikhonov parameter ensuring this independence. We also derive generalizations of these results when the control variable is discretized or when it is taken from a convex set
Time-Parallel Iterative Solvers for Parabolic Evolution Equations
We present original time-parallel algorithms for the solution of the implicit Euler
discretization of general linear parabolic evolution equations with time-dependent self-adjoint spatial operators. Motivated by the inf-sup theory of parabolic problems, we show that the standard
nonsymmetric time-global system can be equivalently reformulated as an original symmetric saddlepoint system that remains inf-sup stable with respect to the same natural parabolic norms. We
then propose and analyse an efficient and readily implementable parallel-in-time preconditioner to
be used with an inexact Uzawa method. The proposed preconditioner is non-intrusive and easy to
implement in practice, and also features the key theoretical advantages of robust spectral bounds,
leading to convergence rates that are independent of the number of time-steps, final time, or spatial
mesh sizes, and also a theoretical parallel complexity that grows only logarithmically with respect to
the number of time-steps. Numerical experiments with large-scale parallel computations show the
effectiveness of the method, along with its good weak and strong scaling properties