LEADER 03344nam 2200445 450 001 9910817388003321 005 20230807203319.0 010 $a3-8325-9159-1 035 $a(CKB)4910000000017340 035 $a(MiAaPQ)EBC5850402 035 $a(Au-PeEL)EBL5850402 035 $a(OCoLC)1112426026 035 $a5a8e86f2-0c08-4611-b188-66c5b0dd2d03 035 $a(EXLCZ)994910000000017340 100 $a20190909d2015 uy 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aAdaptive wavelet methods for variational formulations of nonlinear elliptic PDEs on Tensor-Product domains /$fRoland Pabel 210 1$aKo?ln :$cLogos Verlag Berlin,$d[2015] 210 4$d©2015 215 $a1 online resource (332 pages) 300 $a"Inaugural -Dissertation zur Erlangung des Doktorgrades, de Mathemematisch-Naturwissenschalichen Fakulta?t, der Universita?t zu Ko?ln, vorgelegt von, Roland Pabel, aus Ko?ln--T.P. verso. 311 $a3-8325-4102-0 330 $aLong description: This thesis is concerned with the numerical solution of boundary value problems (BVPs) governed by nonlinear elliptic partial differential equations (PDEs). To iteratively solve such BVPs, it is of primal importance to develop efficient schemes that guarantee convergence of the numerically approximated PDE solutions towards the exact solution. The new adaptive wavelet theory guarantees convergence of adaptive schemes with fixed approximation rates. Furthermore, optimal, i.e., linear, complexity estimates of such adaptive solution methods have been established. These achievements are possible since wavelets allow for a completely new perspective to attack BVPs: namely, to represent PDEs in their original infinite dimensional realm. Wavelets in this context represent function bases with special analytical properties, e.g., the wavelets considered herein are piecewise polynomials, have compact support and norm equivalences between certain function spaces and the ellâ‚‚ sequence spaces of expansion coefficients exist. This theoretical framework is implemented in the course of this thesis in a truly dimensionally unrestricted adaptive wavelet program code, which allows one to harness the proven theoretical results for the first time when numerically solving the above mentioned BVPs. Numerical studies of 2D and 3D PDEs and BVPs demonstrate the feasibility and performance of the developed schemes. The BVPs are solved using an adaptive Uzawa algorithm, which requires repeated solution of nonlinear PDE sub-problems. This thesis presents for the first time a numerically competitive implementation of a new theoretical paradigm to solve nonlinear elliptic PDEs in arbitrary space dimensions with a complete convergence and complexity theory. 606 $aTensor products 606 $aEvolution equations, Nonlinear 615 0$aTensor products. 615 0$aEvolution equations, Nonlinear. 676 $a515.353 700 $aPabel$b Roland$01657887 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910817388003321 996 $aAdaptive wavelet methods for variational formulations of nonlinear elliptic PDEs on Tensor-Product domains$94011583 997 $aUNINA