12,871 research outputs found
On the Convergence of Adaptive Iterative Linearized Galerkin Methods
A wide variety of different (fixed-point) iterative methods for the solution
of nonlinear equations exists. In this work we will revisit a unified iteration
scheme in Hilbert spaces from our previous work that covers some prominent
procedures (including the Zarantonello, Ka\v{c}anov and Newton iteration
methods). In combination with appropriate discretization methods so-called
(adaptive) iterative linearized Galerkin (ILG) schemes are obtained. The main
purpose of this paper is the derivation of an abstract convergence theory for
the unified ILG approach (based on general adaptive Galerkin discretization
methods) proposed in our previous work. The theoretical results will be tested
and compared for the aforementioned three iterative linearization schemes in
the context of adaptive finite element discretizations of strongly monotone
stationary conservation laws
Oscillation-free method for semilinear diffusion equations under noisy initial conditions
Noise in initial conditions from measurement errors can create unwanted
oscillations which propagate in numerical solutions. We present a technique of
prohibiting such oscillation errors when solving initial-boundary-value
problems of semilinear diffusion equations. Symmetric Strang splitting is
applied to the equation for solving the linear diffusion and nonlinear
remainder separately. An oscillation-free scheme is developed for overcoming
any oscillatory behavior when numerically solving the linear diffusion portion.
To demonstrate the ills of stable oscillations, we compare our method using a
weighted implicit Euler scheme to the Crank-Nicolson method. The
oscillation-free feature and stability of our method are analyzed through a
local linearization. The accuracy of our oscillation-free method is proved and
its usefulness is further verified through solving a Fisher-type equation where
oscillation-free solutions are successfully produced in spite of random errors
in the initial conditions.Comment: 19 pages, 9 figure
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