Computing semigroups with error control
Abstract
We develop an algorithm that computes strongly continuous semigroups on infinitedimensional Hilbert spaces with explicit error control. Given a generator $A$, a time $t>0$, an arbitrary initial vector $u_0$ and an error tolerance $\epsilon>0$, the algorithm computes $\exp(tA)u_0$ with error bounded by $\epsilon$. The algorithm is based on a combination of a regularized functional calculus, suitable contour quadrature rules, and the adaptive computation of resolvents in infinite dimensions. As a particular case, we show that it is possible, even when only allowing pointwise evaluation of coefficients, to compute, with error control, semigroups on the unbounded domain $L^2(\mathbb{R}^d)$ that are generated by partial differential operators with polynomially bounded coefficients of locally bounded total variation. For analytic semigroups (and more general Laplace transform inversion), we provide a quadrature rule whose error decreases like $\exp(cN/\log(N))$ for $N$ quadrature points, that remains stable as $N\rightarrow\infty$, and which is also suitable for infinitedimensional operators. Numerical examples are given, including: Schrödinger and wave equations on the aperiodic AmmannBeenker tiling, complex perturbed fractional diffusion equations on $L^2(\mathbb{R})$, and damped EulerBernoulli beam equations.
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 arXiv:
 arXiv:2110.06350
 Bibcode:
 2021arXiv211006350C
 Keywords:

 Mathematics  Numerical Analysis;
 65J08;
 65M15;
 65N35;
 47A10;
 46N40