Numerical Methods for Nonlinear Elliptic Differential Equations

• Applicant: Professor Dr. Klaus Böhmer (†)
• Subject Area: Mathematics
• Term: from 2012 to 2013
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 231694594

The convergence for the bifurcation and center manifold relevant data for all up-to date discretization methods and for nonlinear elliptic equations and parabolic equations is the core in the planned Volume II and has to be proved, cf. K. Böhmer, Numerical Methods for Nonlinear Elliptic Differential a Synopsis. Oxford University Press, Oxford, 772 pp., 2010and Chapters 9,10 in Volume II. In fact, the space discretization of the elliptic part is the dominant problem. A key condition is transferring the equivariance from the original to its discrete (bifurcation) problem, cf. Chapters 10, 11 in the planned Volume II. With the co-authors in USA and Canada of the Chapters 11, 22, 23 in Volume II the essential gaps have to be closed. Additionally with specialists my chosen concepts and a mini symposium at the MAFLAP 2013 with S. Brenner have to be discussed. Volume II will be finished in 2013 and be published again in OUP. For "Numerical Exploitation of Equivariance for Finite Groups" in Chapter 11 in Volume II we have to show that mesh free methods fit into the framework there (cooperation with Allgower, Mei, Fasshauer, Tausch).The convergence results for meshfree methods for nonlinear problems have been recently proved for the first time in a paper with Schaback to apear soon in JCAM. "Numerical Center Manifold Methods" have to determine the local dynamics by combining space and time (full) discretization, Chapter 21 in Volume II. This convergence for full discretization methods has been a totally open question for partial differential equations, except K. Böhmer: On hybrid methods for bifurcation studies for general operator equations. In B. Fiedler, editor, Ergodic theory, Analysiss, and Efficient Simulation of Dynamical Systems, pages 73-107, Berlin, Heidelberg, New York, 2001. Springer. We have to verify that the conditions in Chapter 21 are satisfied for the Boussinesq approximation for the heated cylinder (cooperation) with G. Lewis in Chapter 22 and the geophysical problem of convection of the magma in the earth mantel in Chapter 23 in Volume II with G. Dangelmayr, C. Geiger (cooperation with Dangelmayr). I will consult for the following areas the specialists Guckenheimer, Golubitsky for bifurcation and center manifolds, Fasshauer, Schumaker for approximation theory and Bank, Bowman, Holst, Keyfitz, Minev, Nigam and Asher for elliptic PDEs and DAEs.

• DFG Programme: Research Grants