Continuation for Nonlinear Elliptic Partial Di erential
نویسنده
چکیده
The Multiquadric Radial Basis Function (MQ) Method is a meshless collocation method with global basis functions. It is known to have exponentional convergence for interpolation problems. We descretize nonlinear elliptic PDEs by the MQ method. This results in modest size systems of nonlinear algebraic equations which can be eeciently continued by standard continuation software such as auto and content. Examples are given of detection of bifurcations in 1D and 2D PDEs. These examples show high accuracy with small number of unknowns, as compared with known results from the literature.
منابع مشابه
Unique continuation along an analytic curve for the elliptic partial differential equations
We consider an elliptic partial di erential operator P (x; @) with analytic coe cients and discuss the unique continuation along an analytic curve. That is, let P (x; @)u = 0 in a simply connected domain R, be an analytic curve and let fxgj2N have an accumulation point. Our main result asserts that if u(x) = 0, j 2 N , then u(x) = 0 for any x 2 . Furthermore we apply such uniqueness to an isotr...
متن کاملGeneralized BSDEs and nonlinear Neumann boundary value problems
We study a new class of backward stochastic dierential equations, which involves the integral with respect to a continuous increasing process. This allows us to give a probabilistic formula for solutions of semilinear partial dierential equations with Neumann boundary condition, where the boundary condition itself is nonlinear. We consider both parabolic and elliptic equations.
متن کاملParallel Newton-Krylov-Schwarz Algorithms for the Transonic Full Potential Equation
We study parallel two-level overlapping Schwarz algorithms for solving nonlinear nite element problems, in particular, for the full potential equation of aerodynamics discretized in two dimensions with bilinear elements. The overall algorithm, Newton-Krylov-Schwarz (NKS), employs an inexact nite-di erence Newton method and a Krylov space iterative method, with a two-level overlapping Schwarz me...
متن کاملSpecial Session 33: Nonlinear Elliptic and Parabolic Problems in Mathematical Sciences
Recent developments of mathematical study for nonlinear PDEs (partial di↵erential equations) provide new ideas and various techniques based on calculus of variations, dynamical systems, asymptotic analysis, qualitative theory etc. In this session we bring together researchers in this research area to present new results for nonlinear parabolic and elliptic equations arising from mathematical sc...
متن کاملSchur complement preconditioning for elliptic systems of partial differential equations
S hur omplement pre onditioning for ellipti systems of partial di erential equations D. Loghin and A. J. Wathen One su essful approa h in the design of solution methods for saddle-point problems requires the eÆ ient solution of the asso iated S hur omplement problem ([26℄, [19℄). In the ase of problems arising from partial di erential equations the fa torization of the symbol of the operator an...
متن کامل