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BL
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| Record Number
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862555
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| Main Entry
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Langtangen, Hans Petter,1962-
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| Title & Author
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Introduction to numerical methods for variational problems /\ Hans Petter Langtangen, Kent-Andre Mardal.
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| Publication Statement
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Cham :: Springer,, 2019.
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| Series Statement
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Texts in computational science and engineering ;; 21
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| Page. NO
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1 online resource
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| ISBN
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3030237885
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: 3030237893
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: 3030237907
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: 9783030237882
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: 9783030237899
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: 9783030237905
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3030237877
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9783030237875
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| Contents
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Intro; Preface; Second Preface; Contents; List of Exercises and Problems; 1 Quick Overview of the Finite Element Method; 2 Function Approximation by Global Functions; 2.1 Approximation of Vectors; 2.1.1 Approximation of Planar Vectors; 2.1.2 Approximation of General Vectors; 2.2 Approximation Principles; 2.2.1 The Least Squares Method; 2.2.2 The Projection (or Galerkin) Method; 2.2.3 Example of Linear Approximation; 2.2.4 Implementation of the Least Squares Method; 2.2.5 Perfect Approximation; 2.2.6 The Regression Method; 2.3 Orthogonal Basis Functions; 2.3.1 Ill-Conditioning
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2.3.2 Fourier Series2.3.3 Orthogonal Basis Functions; 2.3.4 Numerical Computations; 2.4 Interpolation; 2.4.1 The Interpolation (or Collocation) Principle; 2.4.2 Lagrange Polynomials; 2.4.3 Bernstein Polynomials; 2.5 Approximation Properties and Convergence Rates; 2.6 Approximation of Functions in Higher Dimensions; 2.6.1 2D Basis Functions as Tensor Products of 1D Functions; 2.6.2 Example on Polynomial Basis in 2D; 2.6.3 Implementation; 2.6.4 Extension to 3D; 2.7 Exercises; Problem 2.1: Linear Algebra Refresher; Problem 2.2: Approximate a Three-Dimensional Vector in a Plane
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3.1.1 Elements and Nodes3.1.2 The Basis Functions; 3.1.3 Example on Quadratic Finite Element Functions; 3.1.4 Example on Linear Finite Element Functions; 3.1.5 Example on Cubic Finite Element Functions; 3.1.6 Calculating the Linear System; 3.1.7 Assembly of Elementwise Computations; 3.1.8 Mapping to a Reference Element; 3.1.9 Example on Integration over a Reference Element; 3.2 Implementation; 3.2.1 Integration; 3.2.2 Linear System Assembly and Solution; 3.2.3 Example on Computing Symbolic Approximations; 3.2.4 Using Interpolation Instead of Least Squares
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3.2.5 Example on Computing Numerical Approximations3.2.6 The Structure of the Coefficient Matrix; 3.2.7 Applications; 3.2.8 Sparse Matrix Storage and Solution; 3.3 Comparison of Finite Elements and Finite Differences; 3.3.1 Finite Difference Approximation of Given Functions; 3.3.2 Interpretation of a Finite Element Approximation in Terms of Finite Difference Operators; 3.3.3 Making Finite Elements Behave as Finite Differences; 3.4 A Generalized Element Concept; 3.4.1 Cells, Vertices, and Degrees of Freedom; 3.4.2 Extended Finite Element Concept; 3.4.3 Implementation
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Problem 2.3: Approximate a Parabola by a SineProblem 2.4: Approximate the Exponential Function by Power Functions; Problem 2.5: Approximate the Sine Function by Power Functions; Problem 2.6: Approximate a Steep Function by Sines; Problem 2.7: Approximate a Steep Function by Sines with Boundary Adjustment; Exercise 2.8: Fourier Series as a Least Squares Approximation; Problem 2.9: Approximate a Steep Function by Lagrange Polynomials; Problem 2.10: Approximate a Steep Function by Lagrange Polynomials and Regression; 3 Function Approximation by Finite Elements; 3.1 Finite Element Basis Functions
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| Abstract
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This textbook teaches finite element methods from a computational point of view. It focuses on how to develop flexible computer programs with Python, a programming language in which a combination of symbolic and numerical tools is used to achieve an explicit and practical derivation of finite element algorithms. The finite element library FEniCS is used throughout the book, but the content is provided in sufficient detail to ensure that students with less mathematical background or mixed programming-language experience will equally benefit. All program examples are available on the Internet.
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| Subject
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Finite element method.
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| Subject
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Variational inequalities (Mathematics)
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| Subject
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Finite element method.
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| Subject
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Variational inequalities (Mathematics)
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| Dewey Classification
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515/.64
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| LC Classification
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QA316
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| Added Entry
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Mardal, Kent-Andre.
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