65J15 Equations with nonlinear operators (do not use 65Hxx)
Refine
Document Type
- ZIB-Report (3)
Language
- English (3)
Has Fulltext
- yes (3)
Is part of the Bibliography
- no (3)
Keywords
- Newton method (1)
- Rothe method (1)
- adaptive Runge-Kutta method (1)
- collocation method (1)
- finite element method (1)
- finite elements (1)
- hyperthermia (1)
- initial boundary value problem (1)
- mesh independence (1)
- mesh refinement (1)
Institute
- ZIB Allgemein (2)
- Computational Medicine (1)
- Numerical Mathematics (1)
The paper presents a new affine invariant theory on asymptotic mesh independence of Newton's method in nonlinear PDEs. Compared to earlier attempts, the new approach is both much simpler and more natural from the algorithmic point of view. The theory is exemplified at collocation methods for ODE boundary value problems and at finite element methods for elliptic PDE problems.
In the clinical cancer therapy of regional hyperthermia nonlinear perfusion effects inside and outside the tumor seem to play a not negligible role. A stationary model of such effects leads to a nonlinear Helmholtz term within an elliptic boundary value problem. The present paper reports about the application of a recently designed adaptive multilevel FEM to this problem. For several 3D virtual patients, nonlinear versus linear model is studied. Moreover, the numerical efficiency of the new algorithm is compared with a former application of an adaptive FEM to the corresponding instationary model PDE.
A Finite Element Method Adaptive in Space and Time for Nonlinear Reaction-Diffusion- Systems.
(1992)
Large scale combustion simulations show the need for adaptive methods. First, to save computation time and mainly to resolve local and instationary phenomena. In contrast to the widespread method of lines, we look at the reaction- diffusion equations as an abstract Cauchy problem in an appropriate Hilbert space. This means, we first discretize in time, assuming the space problems solved up to a prescribed tolerance. So, we are able to control the space and time error separately in an adaptive approach. The time discretization is done by several adaptive Runge-Kutta methods whereas for the space discretization a finite element method is used. The different behaviour of the proposed approaches are demonstrated on many fundamental examples from ecology, flame propagation, electrodynamics and combustion theory. {\bf Keywords:} initial boundary value problem, Rothe- method, adaptive Runge-Kutta method, finite elements, mesh refinement. {\bf AMS CLASSIFICATION:} 65J15, 65M30, 65M50.