Self-Adaptive Methods for PDE

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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

Modelling of the advection-diffusion equation with a meshless method without numerical diffusion

A comprehensive study is presented regarding the stability of the forward explicit integration technique with generalized finite difference spatial discretizations, free of numerical diffusion, applied to the advection-diffusion equation. The modified equivalent partial differential equation approach is used to demonstrate that the approximation is free of numerical diffusion. Two-dimensional results are obtained using the von Neumann method of stability analysis. Numerical results are presented showing the accuracy obtaine

Adaptive finite element methods for linear-quadratic convection dominated elliptic optimal control problems

The numerical solution of linear-quadratic elliptic optimal control problems requires the solution of a coupled system of elliptic partial differential equations (PDEs), consisting of the so-called state PDE, the adjoint PDE and an algebraic equation. Adaptive finite element methods (AFEMs) attempt to locally refine a base mesh in such a way that the solution error is minimized for a given discretization size. This is particularly important for the solution of convection dominated problems where inner and boundary layers in the solutions to the PDEs need to be sufficiently resolved to ensure that the solution of the discretized optimal control problem is a good approximation of the true solution. This thesis reviews several AFEMs based on energy norm based error estimates for single convection dominated PDEs and extends them to the solution of the coupled system of convection dominated PDEs arising from the optimality conditions for optimal control problems. Keywords Adaptive finite element methods, optimal control problems, convection-diffusion equations, local refinement, error estimation

Reaction-diffusion systems on evolving domains with applications to the theory of biological pattern formation

In this thesis we investigate a model for biological pattern formation during growth development. The pattern formation phenomenon is described by a reaction-diffusion system on a time-dependent domain. We prove the global existence of solutions to reaction-diffusion systems on time-dependent domains.We extend global existence results for a class of reaction-diffusion systems on fixed domains to the same systems posed on spatially linear isotropically evolving domains. We demonstrate that the analysis is applicable to many systems that commonly arise in the theory of pattern formation. Our results give a mathematical justification to the widespread use of computer simulations of reaction-diffusion systems on evolving domains. We propose a finite element method to approximate the solutions to reaction-diffusion systems on time-dependent domains. We prove optimal convergence rates for the error in the method and we derive a computable error estimator that provides an upper bound for the error in the semidiscrete (space) scheme. We have implemented the method in the C programming language and we verify our theoretical results with benchmark computations. The method is a robust tool for the study of biological pattern formation, as it is applicable to domains with irregular geometries and nonuniform evolution. This versatility is illustrated with extensive computer simulations of reaction-diffusion systems on evolving domains. We observe varied pattern transitions induced by domain evolution, such as stripe to spot transitions, spotsplitting, spot-merging and spot-annihilation. We also illustrate the striking effects of spatially nonuniform domain evolution on the position, orientation and symmetry of patterns generated by reaction-diffusion systems. To improve the efficiency of the method, we have implemented a space-time adaptive algorithm where spatial adaptivity is driven by an error estimator and temporal adaptivity is driven by an error indicator.We illustrate with numerical simulations the dramatic improvements in accuracy and efficiency that are achieved via adaptivity. To demonstrate the applicability and generality of our methodology, we examine the process of parr mark pattern formation during the early development of the Amago trout. By assuming the existence of chemical concentrations residing on the surface of the Amago fish which react and diffuse during surface evolution, we model the pattern formation process with reactiondiffusion systems posed on evolving surfaces. An important generalisation of our study is the experimentally driven modelling of the fish’s developing body surface. Our results add weight to the feasibility of reaction-diffusion system models of fish skin patterning, by illustrating that a reaction-diffusion system posed on an evolving surface generates transient patterns consistent with those experimentally observed on the developing Amago trout. Furthermore, we conclude that the surface evolution profile, the surface geometry and the curvature are key factors which play a pivotal role in pattern formation via reaction-diffusion systems

Error estimation for simplifications of electrostatic models

Based on a posteriori error estimation a method to bound the error induced by simplifying the geometry of a model is presented. Error here refers to the solution of a partial differential equation and a specific quantity of interest derived from it. Geometry simplification specifically refers to replacing CAD model features with simpler shapes. The simplification error estimate helps to determine whether a feature can be removed from the model by indicating how much the simplification affects the physical properties of the model as measured by a quantity of interest. The approach in general can also be extended to other problems governed by linear elliptic equations. Strict bounds for the error are proven for errors expressed in the energy norm. The approach relies on the Constitutive Relation Error to enable practically useful and computationally affordable bounds for error measures in the energy error norm. All methodologies are demonstrated for a second order elliptic partial differential equation for electrostatic problems. Finite element simplification error estimation code is developed to calculate the simplification error numerically. Numerical experiments for some geometric models of capacitors show satisfactory results for the simplification error bounds for a range of different deafeaturing cases and a quantity of interest, linear in the solution of the electrostatic partial differential equation. Overall the numerically calculated bounds are always valid, but are more or less accurate depending on the type of feature and its simplification. In particular larger errors may be overestimated, while good estimates for small errors can be achieved. This makes the bound overall suitable to decide whether simplifying a feature is acceptable or not