381 research outputs found
The DPG-star method
This article introduces the DPG-star (from now on, denoted DPG) finite
element method. It is a method that is in some sense dual to the discontinuous
Petrov-Galerkin (DPG) method. The DPG methodology can be viewed as a means to
solve an overdetermined discretization of a boundary value problem. In the same
vein, the DPG methodology is a means to solve an underdetermined
discretization. These two viewpoints are developed by embedding the same
operator equation into two different saddle-point problems. The analyses of the
two problems have many common elements. Comparison to other methods in the
literature round out the newly garnered perspective. Notably, DPG and DPG
methods can be seen as generalizations of and
least-squares methods, respectively. A priori error analysis and a posteriori
error control for the DPG method are considered in detail. Reports of
several numerical experiments are provided which demonstrate the essential
features of the new method. A notable difference between the results from the
DPG and DPG analyses is that the convergence rates of the former are
limited by the regularity of an extraneous Lagrange multiplier variable
Adaptive FEM for parameter-errors in elliptic linear-quadratic parameter estimation problems
We consider an elliptic linear-quadratic parameter estimation problem with a
finite number of parameters. A novel a priori bound for the parameter error is
proved and, based on this bound, an adaptive finite element method driven by an
a posteriori error estimator is presented. Unlike prior results in the
literature, our estimator, which is composed of standard energy error residual
estimators for the state equation and suitable co-state problems, reflects the
faster convergence of the parameter error compared to the (co)-state variables.
We show optimal convergence rates of our method; in particular and unlike prior
works, we prove that the estimator decreases with a rate that is the sum of the
best approximation rates of the state and co-state variables. Experiments
confirm that our method matches the convergence rate of the parameter error
Self-Adaptive Methods for PDE
[no abstract available
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