Sesión Análisis Numérico y OptimizaciónHow much information can we recover from piecewise polynomial approximations?
Thomas Fuehrer
Pontificia Universidad Católica de Chile, Chile - Esta dirección de correo electrónico está siendo protegida contra los robots de spam. Necesita tener JavaScript habilitado para poder verlo.
It is well known that the $L^2$ best approximation $\Pi_\mathcal{T}^p$ onto $\mathcal{T}$-piecewise polynomials of degree $\leq p\in\mathbb{N}_0$ satisfies \[ \|u-\Pi_\mathcal{T}^p u\|_{L^2(\Omega)} \lesssim h^{p+1}\|D^{p+1}u\|_{L^2(\Omega)} \] and this result is sharp. Here, $\mathcal{T}$ is a regular simplicial mesh of the Lipschtiz domain $\Omega\subseteq \mathbb{R}^d$ ($d=1,2,3$). In this talk I present recent results from [1] where we show how to recover approximations from $\mathcal{T}$-piecewise polynomial approximation that converge at higher rates. We do this by introducing a family of quasi-interpolation operators $J^p$ that satisfy $J^p = J^p \circ \Pi_\mathcal{T}^p$, i.e., the operators only ``see'' piecewise polynomials of degree $p$, and that map into the space of piecewise polynomials of degree $p+1$. We prove for $d=1,2$ that these operators satisfy \[ \|u-J^p u\|_{L^2(\Omega)} \lesssim h^{p+2}\|D^{p+2}u\|_{L^2(\Omega)}. \] This means that $J^p u$ converges at a higher rate than the best approximation $\Pi_\mathcal{T}^p u$ although both use the same information. While the latter result is of purely theoretic nature, it can be exploited to enhance accuracy of, e.g., finite element solutions $u_h$. Indeed, many methods including mixed finite element methods, (hybridizable) discontinuous (Petrov-)Galerkin methods satisfy a supercloseness property of the form (or similar) \[ \|\Pi_\mathcal{T}^p(u-u_h)\|_{L^2(\Omega)} = \mathcal{O}(h^{p+2}). \] Combining this observation with our operator we prove that \[ \|u-J^pu_h\|_{L^2(\Omega)} =\mathcal{O}(h^{p+2}), \qquad\text{although } \|u-u_h\|_{L^2(\Omega)} = \mathcal{O}(h^{p+1}). \] While popular postprocessing like [2] achieve the same order of convergence and can be computed more efficiently, our approach extends to more general situations where, e.g., discrete gradient approximations are not directly accesible.
A second family of operators is introduced that is based on piecewise constant approximations and for $d=1$ we establish that an arbitrary (but fixed) order of convergence can be achieved under mild assumptions on the mesh. Throughout the talk we present numerical examples.
Our main results are based on highly technical results that analyze the intersection of orthogonal polynomials on patches [3].
Trabajo en conjunto con: Manuel A. Sánchez (Pontificia Universidad Católica de Chile).
Referencias
[1] T. Führer and M. A. Sánchez. Quasi-interpolators with application to postprocessing in finite element methods. preprint, arXiv:2404.13183, 2024.
[2] R. Stenberg. Postprocessing schemes for some mixed finite elements. RAIRO Modélisation Mathématique et Analyse Numérique, 25(1):151-167, 1991.
[3] T.H. Koornwinder and S. Sauter. The intersection of bivariate orthogonal polynomials on triangle patches. Mathematics of Computation, 84(294):1795-1812, 2015.