With in-depth refinement, the condition number of the incremental unknowns matrix associated to the Laplace operator is $p(d)O(1/H^2)O(|log_d h|^3)$ for the first order incremental unknowns, and $q(d)O(1/H^2)O((log_d h)^2)$ for the second order incremental unknowns, where $d$ is the depth of the refinement, $H$ is the mesh size of the coarsest grid, $h$ is the mesh size of the finest grid, $p(d) = \frac{d-1}{2}$ and $q(d) = \frac{d-1}{2} \frac{1}{12}d(d^2-1)$. Furthermore, if block diagonal (scaling) preconditioning is used, the condition number of the preconditioned incremental unknowns matrix associated to the Laplace operator is $p(d) O((log_d h)^2)$ for the first order incremental unknowns, and $q(d)O(|log_dh|)$ for the second order incremental unknowns. For comparison, the condition number of the nodal unknowns matrix associated to the Laplace operator is $O(1/h^2)$. Therefore, the incremental unknowns preconditioner is efficient with in-depth refinement, but its efficiency deteriorates at some rate as the depth of the refinement grows.

The well-known logistic model has been extensively investigated in deterministic theory. There are numerous case studies where such type of nonlinearities occur in Ecology, Biology and Environmental Sciences. Due to the presence of environmental fluctuations and a lack of precision of measurements, one has to deal with effects of randomness on such models. As a more realistic modeling, we suggest nonlinear stochastic differential equations (SDEs) $$dX(t) = [(\rho + \lambda X(t))(K - X(t)) - \mu X(t)]dt + \sigma X(t)^{\alpha}| K - X(t)|^{\beta}dW(t)$$ of Itô type to model the growth of populations or innovations $X$, driven by a Wiener process $W$ and positive real constants $\rho$, $\lambda$, $K$, $\mu$, $\alpha$, $\beta \geq 0$. We discuss well-posedness, regularity (boundedness) and uniqueness of their solutions. However, explicit expressions for analytical solution of such random logistic equations are rarely known. Therefore one has to resort to numerical solution of SDEs for studying various aspects like the time-evolution of growth patterns, exit frequencies, mean passage times and impact of fluctuating growth parameters. We present some basic aspects of adequate numerical analysis of these random extensions of these models such as numerical regularity and mean square convergence. The problem of keeping reasonable boundaries for analytic solutions under discretization plays an essential role for practically meaningful models, in particular the preservation of intervals with reflecting or absorbing barriers. A discretization of the continuous state space can be circumvented by appropriate methods. Balanced implicit methods (see Schurz, IJNAM 2 (2), pp. 197-220, 2005) are used to construct strongly converging approximations with the desired monotone properties. Numerical studies can bring out salient features of the stochastic logistic models (e.g. almost sure monotonicity, almost sure uniform boundedness, delayed initial evolution or earlier points of inflection compared to deterministic model).

In this paper, a set of new alternating segment explicit-implicit (NASEI) schemes is derived based on a one-dimensional diffusion problem. The schemes are capable of parallel computation; third-order accurate in space; and stable under a reasonable mesh condition. The numerical examples show that the NASEI schemes are more accurate than either the old ASEI or the ASCN schemes.

Implicit integration schemes for ODEs, such as Runge-Kutta and Runge-Kutta-Nyström methods, are widely used in mathematics and engineering to numerically solve ordinary differential equations. Every integration method requires one to choose a step-size, $h$, for the integration. If $h$ is too large or too small the efficiency of an implicit scheme is relatively low. As every implicit integration scheme has a global error inherent to the scheme, we choose the total number of computations in order to achieve a prescribed global error as a measure of efficiency of the integration scheme. In this paper, we propose the idea of choosing $h$ by minimizing an efficiency function for general Runge-Kutta and Runge-Kutta-Nyström integration routines. This efficiency function is the critical component in making these methods variable step-size methods. We also investigate solving the intermediate stage values of these routines using both Newton's method and Picard iteration. We then show the efficacy of this approach on some standard problems found in the literature, including a well-known stiff system.

The author establishes a finite element solver algorithm of optimal speed for a class of quasi-linear equations with large stiffness variations and oscillations. In particular, the algorithm can successfully handle soft inclusions of negative stiffness. Besides the convergence analysis, large number of numerical examples are presented.

General multi-dimensional autonomous dynamical systems and their numerical discretizations are considered. Nonstandard stability-preserving finite-difference schemes based on the $\theta$-methods and the second-order Runge-Kutta methods are designed and analyzed. Their elementary stability is established theoretically and is also supported by a set of numerical examples.

We apply the Piecewise Constant Level Set Method (PCLSM) to a multiphase motion problem, especially the pure mean curvature motion. We use one level set function to represent multiple regions, and by associating an energy functional which consists of surface tension (proportional to length), we formulate a variational approach for the mean curvature motion problem. Some operator-splitting schemes are used to solve the problem efficiently. Numerical experiments are supplied to show the efficiency for motion by mean curvature for multiphase problems.