Robust Numerical Methods for Singularly Perturbed Differential Equations--Supplements

The second edition of the book "Roos, Stynes, Tobiska -- Robust Numerical Methods for Singularly Perturbed Differential Equations" appeared many years ago and was for many years a reliable guide into the world of numerical methods for singularly perturbed problems. Since then many new results came into the game, we present some selected ones and the related sources.Comment: arXiv admin note: text overlap with arXiv:1909.0827

An hphp-Adaptive Newton-Galerkin Finite Element Procedure for Semilinear Boundary Value Problems

In this paper we develop an $hp$-adaptive procedure for the numerical solution of general, semilinear elliptic boundary value problems in 1d, with possible singular perturbations. Our approach combines both a prediction-type adaptive Newton method and an $hp$-version adaptive finite element discretization (based on a robust a posteriori residual analysis), thereby leading to a fully $hp$-adaptive Newton-Galerkin scheme. Numerical experiments underline the robustness and reliability of the proposed approach for various examples.Comment: arXiv admin note: text overlap with arXiv:1408.522

Uniformly convergent additive schemes for 2d singularly perturbed parabolic systems of reaction-diffusion type

In this work, we consider parabolic 2D singularly perturbed systems of reaction-diffusion type on a rectangle, in the simplest case that the diffusion parameter is the same for all equations of the system. The solution is approximated on a Shishkin mesh with two splitting or additive methods in time and standard central differences in space. It is proved that they are first-order in time and almost second-order in space uniformly convergent schemes. The additive schemes decouple the components of the vector solution at each time level of the discretization which makes the computation more efficient. Moreover, a multigrid algorithm is used to solve the resulting linear systems. Numerical results for some test problems are showed, which illustrate the theoretical results and the efficiency of the splitting and multigrid techniques

Landscapes of Non-gradient Dynamics Without Detailed Balance: Stable Limit Cycles and Multiple Attractors

Landscape is one of the key notions in literature on biological processes and physics of complex systems with both deterministic and stochastic dynamics. The large deviation theory (LDT) provides a possible mathematical basis for the scientists' intuition. In terms of Freidlin-Wentzell's LDT, we discuss explicitly two issues in singularly perturbed stationary diffusion processes arisen from nonlinear differential equations: (1) For a process whose corresponding ordinary differential equation has a stable limit cycle, the stationary solution exhibits a clear separation of time scales: an exponential terms and an algebraic prefactor. The large deviation rate function attains its minimum zero on the entire stable limit cycle, while the leading term of the prefactor is inversely proportional to the velocity of the non-uniform periodic oscillation on the cycle. (2) For dynamics with multiple stable fixed points and saddles, there is in general a breakdown of detailed balance among the corresponding attractors. Two landscapes, a local and a global, arise in LDT, and a Markov jumping process with cycle flux emerges in the low-noise limit. A local landscape is pertinent to the transition rates between neighboring stable fixed points; and the global landscape defines a nonequilibrium steady state. There would be nondifferentiable points in the latter for a stationary dynamics with cycle flux. LDT serving as the mathematical foundation for emergent landscapes deserves further investigations.Comment: 4 figur