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Classical Numerical Methods in Scientific Computing

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Authors

Guus Segal

Delft University of Technology, Faculty of Electrical Engineering, Mathematics and Computer Science, Department of Applied Mathematics

Keywords:
partial differential equations, finite element method, discretization, time integration, mathematical modelling

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Synopsis

Partial differential equations are paramount in mathematical modelling with applications in engineering and science. The book starts with a crash course on partial differential equations in order to familiarize the reader with fundamental properties such as existence, uniqueness and possibly existing maximum principles. The main topic of the book entails the description of classical numerical methods that are used to approximate the solution of partial differential equations. The focus is on discretization methods such as the finite difference, finite volume and finite element method. The manuscript also makes a short excursion to the solution of large sets of (non)linear algebraic equations that result after application of discretization method to partial differential equations. The book treats the construction of such discretization methods, as well as some error analysis, where it is noted that the error analysis for the finite element method is merely descriptive, rather than rigorous from a mathematical point of view. The last chapters focus on time integration issues for classical time-dependent partial differential equations. After reading the book, the reader should be able to derive finite element methods, to implement the methods and to judge whether the obtained approximations are consistent with the solution to the partial differential equations. The reader will also obtain these skills for the other classical discretization methods. Acquiring such fundamental knowledge will allow the reader to continue studying more advanced methods like meshfree methods, discontinuous Galerkin methods and spectral methods for the approximation of solutions to partial differential equations.

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Author Biographies

Jos van Kan, Delft University of Technology, Faculty of Electrical Engineering, Mathematics and Computer Science, Department of Applied Mathematics

Jos van Kan (1944) graduated in 1968 from Delft University of Technology, Delft, Netherlands, in Numerical Analysis and was assistant professor at the Department of Mathematics of that institute until 2009. He wrote several articles on Numerical Fluid Mechanics (pressure correction methods) and has written a multigrid pressure solver for the Delft software package to solve the Navier-Stokes equations. He was teaching classes in Numerical Analysis from 1971 until 2009, and wrote several books on the subject. Currently he is a retired professor.

Guus Segal, Delft University of Technology, Faculty of Electrical Engineering, Mathematics and Computer Science, Department of Applied Mathematics

Guus Segal (1948) graduated in 1971 from Delft University of Technology, Delft, Netherlands, in Numerical Analysis and was part time assistant professor at the Department of Mathematics of that institute until 2013. He also worked in the consultancy and numerical software company SEPRA in The Hague, Netherlands. He wrote a number of articles on Finite Element Methods and several articles on curvilinear Finite Volume Methods and Numerical Fluid Mechanics. He has written a book on Finite Element methods and Navier-Stokes equations. He is the main developer of the finite element package SEPRAN. He was teaching classes in Numerical Analysis from 1973 until 2013.

Fred Vermolen, University of Hasselt, Faculty of Science, Department of Computational Mathematics

Fred Vermolen (1969) graduated in 1993 from Delft University of Technology, Delft, Netherlands and defended his PhD thesis on numerical methods for moving boundary problems in 1998. He has written several contributions on Stefan problems, computational mechanics, mathematical analysis and uncertainty quantification with most of the applications in medicine. He has held an assistant and associate professorship in Numerical Analysis at the Delft University from 2000 until 2020. In 2020 he started his current position as a full professor in Computational Mathematics at the University of Hasselt in Belgium.

Copyright (c) 2023 Jos van Kan, Guus Segal, Fred Vermolen