Computer-assisted Existence Proofs for Navier-Stokes Equations on an Unbounded Strip with Obstacle

The incompressible stationary 2D Navier-Stokes equations are considered on an unbounded strip domain with a compact obstacle. First, a computer-assisted existence and enclosure result for the velocity (in a suitable divergence-free Sobolev space) is presented. Starting from an approximate solution (computed with divergence-free finite elements), we determine a bound for its defect, and a norm bound for the inverse of the linearization at the approximate solution. For the latter, bounds for the essential spectrum and for eigenvalues play a crucial role, especially for the eigenvalues ``close to\u27\u27 zero. Note that, on an unbounded domain, the only general method for computing the desired norm bound appears to be via eigenvalue bounds. To obtain the desired lower bounds for the eigenvalues below the essential spectrum we use the Rayleigh-Ritz method, a corollary of the Temple-Lehmann theorem and a homotopy method. Finally, if the computer-assisted proof provides the existence of a velocity field, the existence of a corresponding pressure can be obtained by purely analytical techniques. Nevertheless, for a given approximate solution to the pressure our methods provide an error bound (in a dual norm) as well

Computer-assisted Multiplicity Proofs for Emden\u27s Equation on Domains with Hole

In this thesis we consider Emden\u27s equation on a parameter dependent domain with hole and describe a computer-assisted method for proving existence and multiplicity of solutions to that problem. We obtain results for discrete parameter values as well as for some parameter intervals, in which case also the existence of smooth solution branches is proved. Moreover we prove the existence of a one-bump solution to Emden\u27s equation on an unbounded L-shaped domain

On the computation of geometric features of spectra of linear operators on Hilbert spaces

Computing spectra is a central problem in computational mathematics with an abundance of applications throughout the sciences. However, in many applications gaining an approximation of the spectrum is not enough. Often it is vital to determine geometric features of spectra such as Lebesgue measure, capacity or fractal dimensions, different types of spectral radii and numerical ranges, or to detect essential spectral gaps and the corresponding failure of the finite section method. Despite new results on computing spectra and the substantial interest in these geometric problems, there remain no general methods able to compute such geometric features of spectra of infinite-dimensional operators. We provide the first algorithms for the computation of many of these longstanding problems (including the above). As demonstrated with computational examples, the new algorithms yield a library of new methods. Recent progress in computational spectral problems in infinite dimensions has led to the Solvability Complexity Index (SCI) hierarchy, which classifies the difficulty of computational problems. These results reveal that infinite-dimensional spectral problems yield an intricate infinite classification theory determining which spectral problems can be solved and with which type of algorithm. This is very much related to S. Smale's comprehensive program on the foundations of computational mathematics initiated in the 1980s. We classify the computation of geometric features of spectra in the SCI hierarchy, allowing us to precisely determine the boundaries of what computers can achieve and prove that our algorithms are optimal. We also provide a new universal technique for establishing lower bounds in the SCI hierarchy, which both greatly simplifies previous SCI arguments and allows new, formerly unattainable, classifications