Aim of the course
Presentation of the various basic aspects of computer simulation methods for continuous computing problems, including sequential and parallel algorithms for a) bringing a computational problem to the form of algebraic systems of linear equations, b) efficient solving of systems of linear equations, and c) adaptive algorithms for solution accuracy control.
Examples of computational problems described by ordinary and partial differential equations: the problem of unsteady heat transport, propagation of electromagnetic and acoustic waves (the problem of identification of mineral resources, or tumor changes, acoustic model of a human head). Differential patterns of discretization with respect to time (simple schemes, mixed schemes, adaptative schemes). Algorithms of generation of system of algebraic linear equations for sample computational issues (finite element method, finite difference method). Sequential and parallel algorithms for solving large systems of linear equations (LU factorization, the frontal algorithm, the multifrontal algorithm). Methods of assessing the accuracy of obtained numerical solution. Sequential and parallel adaptive algorithms. Examples of applications of adaptive algorithms (adaptive differential schemes, adaptive finite element method, adaptive image compression.)
Overview of the course elements
The course involves laboratory classes. The content of the course consolidates and extends the knowledge taught at the lecture. As part of the laboratory projects students will implement application projects for selected aspects of the algorithms presented at the lecture. Sample tasks may include design and implementation of the algorithm of multi-frontal solver in a parallel environment, design and implementation of sequential or parallel adaptive algorithms for solving various computational problems, design and implementation of algorithms for implementing the mixed schemes of solving non-stationary problem, design and implementation of adaptive algorithms for performing image compression, etc.
1. Gene H. Golub, Charles E. Van Loan; Matrix computations. The John Hopkins University Press, Third edition 1996
2. Demkowicz L. Computing with hp-Adaptive Finite Elements. Vol. 1: One and Two Dimensional Elliptic and Maxwell Problems, Chapmann & Hall / CRC Press 2006
3. Demkowicz L. Kurtz J., Pardo D., Paszyński M., Rachowicz W., Zdunek A., Computing with hp- Adaptive Finite Elements. Vol. 2: Frontiers: Three Dimensional Elliptic and Maxwell Problems with Applications, Chapmann & Hall / CRC Press 2007
4. Hughes T. J. R. The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Dover Publications, 2000
5. Paszyński M., Graph grammar driven parallel adaptive PDE solvers, AGH Uczelnianie Wydawnictwa Naukowe 2009