Optimal L∞(L2)L^\infty(L^2) and L1(L2)L^1(L^2) a posteriori error estimates for the fully discrete approximations of time fractional parabolic differential equations

We derive optimal order a posteriori error estimates in the $L^\infty(L^2)$ and $L^1(L^2)$-norms for the fully discrete approximations of time fractional parabolic differential equations. For the discretization in time, we use the $L1$ methods, while for the spatial discretization, we use standard conforming finite element methods. The linear and quadratic space-time reconstructions are introduced, which are generalizations of the elliptic space reconstruction. Then the related a posteriori error estimates for the linear and quadratic space-time reconstructions play key roles in deriving global and pointwise final error estimates. Numerical experiments verify and complement our theoretical results.Comment: 22 page

Computationally efficient solution to the Cahn–Hilliard equation: Adaptive implicit time schemes, mesh sensitivity analysis and the 3D isoperimetric problem

We present an efficient numerical framework for analyzing spinodal decomposition described by the Cahn–Hilliard equation. We focus on the analysis of various implicit time schemes for two and three dimensional problems. We demonstrate that significant computational gains can be obtained by applying embedded, higher order Runge–Kutta methods in a time adaptive setting. This allows accessing time-scales that vary by five orders of magnitude. In addition, we also formulate a set of test problems that isolate each of the sub-processes involved in spinodal decomposition: interface creation and bulky phase coarsening. We analyze the error fluctuations using these test problems on the split form of the Cahn–Hilliard equation solved using the finite element method with basis functions of different orders. Any scheme that ensures at least four elements per interface satisfactorily captures both sub-processes. Our findings show that linear basis functions have superior error-to-cost properties. This strategy – coupled with a domain decomposition based parallel implementation – let us notably augment the efficiency of a numerical Cahn–Hillard solver, and open new venues for its practical applications, especially when three dimensional problems are considered. We use this framework to address the isoperimetric problem of identifying local solutions in the periodic cube in three dimensions. The framework is able to generate all five hypothesized candidates for the local solution of periodic isoperimetric problem in 3D – sphere, cylinder, lamella, doubly periodic surface with genus two (Lawson surface) and triply periodic minimal surface (P Schwarz surface)

Snapshot-Based Methods and Algorithms

An increasing complexity of models used to predict real-world systems leads to the need for algorithms to replace complex models with far simpler ones, while preserving the accuracy of the predictions. This two-volume handbook covers methods as well as applications. This second volume focuses on applications in engineering, biomedical engineering, computational physics and computer science

Discontinuous Galerkin Method Applied to Navier-Stokes Equations

Discontinuous Galerkin (DG) finite element methods are becoming important techniques for the computational solution of many real-world problems describe by differential equations. They combine many attractive features of the finite element and the finite volume methods. These methods have been successfully applied to many important PDEs arising from a wide range of applications. DG methods are highly accurate numerical methods and have considerable advantages over the classical numerical methods available in the literature. DG methods can easily handle meshes with hanging nodes, elements of various types and shapes, and local spaces of different orders. Furthermore, DG methods provide accurate and efficient simulation of physical and engineering problems, especially in settings where the solutions exhibit poor regularity. For these reasons, they have attracted the attention of many researchers working in diverse areas, from computational fluid dynamics, solid mechanics and optimal control, to finance, biology and geology. In this talk, we give an overview of the main features of DG methods and their extensions. We first introduce the DG method for solving classical differential equations. Then, we extend the methods to other equations such as Navier-Stokes equations. The Navier-Stokes equations are useful because they describe the physics of many phenomena of scientific and engineering interest. They may be used to model the weather, ocean currents, water flow in a pipe and air flow around a wing

Model Order Reduction

An increasing complexity of models used to predict real-world systems leads to the need for algorithms to replace complex models with far simpler ones, while preserving the accuracy of the predictions. This two-volume handbook covers methods as well as applications. This second volume focuses on applications in engineering, biomedical engineering, computational physics and computer science

Numerical Methods for Stochastic Allen-Cahn Equation and Stochastic Subdiffusion and Superdiffusion

In this Thesis, we consider the numerical solution of stochastic partial differential equations with particular interest on the Ɛ-dependent Allen-Cahn equation, and the stochastic time fractional partial differential equations in both subdiffusion and superdiffusion cases

Self-Evaluation Applied Mathematics 2003-2008 University of Twente

This report contains the self-study for the research assessment of the Department of Applied Mathematics (AM) of the Faculty of Electrical Engineering, Mathematics and Computer Science (EEMCS) at the University of Twente (UT). The report provides the information for the Research Assessment Committee for Applied Mathematics, dealing with mathematical sciences at the three universities of technology in the Netherlands. It describes the state of affairs pertaining to the period 1 January 2003 to 31 December 2008