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PETSc for Partial Differential Equations: Numerical Solutions in C and Python
  • 2023-08-27
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    ###-Book Description Begin-###
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    PETSc for Partial Differential Equations: Numerical Solutions in C and Python

    PETSc for Partial Differential Equations Numerical Solutions in C and Python.jpg

    Author(s): Ed Bueler

    Ed Bueler

    University of Alaska Fairbanks, Fairbanks, Alaska, USA

    https://doi.org/10.1137/1.9781611976311

    Permalink: https://doi.org/10.1137/1.9781611976311

    Keywords: Partial differential equationsScientific computingPreconditioningSupercomputingSupercomputersNumerical analysis

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    The Portable, Extensible Toolkit for Scientific Computation (PETSc) is an open-source library of advanced data structures and methods for solving linear and nonlinear equations and for managing discretizations. This book uses these modern numerical tools to demonstrate how to solve nonlinear partial differential equations (PDEs) in parallel. It starts from key mathematical concepts, such as Krylov space methods, preconditioning, multigrid, and Newton's method. In PETSc these components are composed at run time into fast solvers.

    Discretizations are introduced from the beginning, with an emphasis on finite difference and finite element methodologies. The example C programs of the first 12 chapters, listed on the inside front cover, solve (mostly) elliptic and parabolic PDE problems. Discretization leads to large, sparse, and generally nonlinear systems of algebraic equations. For such problems, mathematical solver concepts are explained and illustrated through the examples, with sufficient context to speed further development.

    PETSc for Partial Differential Equations

    • addresses both discretization and fast solvers for PDEs;

    • emphasizes practice more than theory;

    • contains well-structured examples, with advice on run-time solver choices;

    • demonstrates how to achieve high performance and parallel scalability; and

    • builds on the reader's understanding of fast solver concepts when applying the Firedrake

    Python finite element solver library in the last two chapters.

    This textbook, the first to cover PETSc programming for nonlinear PDEs, provides an on-ramp for graduate students and researchers to a major area of high-performance computing for science and engineering. It is suitable as a supplement for courses in scientific computing or numerical methods for differential equations.

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    Free Access

    Front Matter

    pp. i-xv (15 pages)

    https://doi.org/10.1137/1.9781611976311.fm

    Abstract | PDF (401 KB) 

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    Chapter 1: Getting started with PETSc

    pp. 1-8 (8 pages)

    https://doi.org/10.1137/1.9781611976311.ch1

    Abstract | PDF (297 KB) 

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    Chapter 2: Finite-dimensional linear systems

    pp. 9-41 (33 pages)

    https://doi.org/10.1137/1.9781611976311.ch2

    Abstract | PDF (603 KB) 

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    Chapter 3: Poisson equation on a structured grid

    pp. 43-66 (24 pages)

    https://doi.org/10.1137/1.9781611976311.ch3

    Abstract | PDF (705 KB) 

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    Chapter 4: Nonlinear equations by Newton's method

    pp. 67-93 (27 pages)

    https://doi.org/10.1137/1.9781611976311.ch4

    Abstract | PDF (620 KB) 

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    Chapter 5: Time-stepping

    pp. 95-128 (34 pages)

    https://doi.org/10.1137/1.9781611976311.ch5

    Abstract | PDF (951 KB) 

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    Chapter 6: Preconditioners for PDEs

    pp. 129-174 (46 pages)

    https://doi.org/10.1137/1.9781611976311.ch6

    Abstract | PDF (777 KB) 

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    Chapter 7: Optimal solvers for elliptic PDEs

    pp. 175-197 (23 pages)

    https://doi.org/10.1137/1.9781611976311.ch7

    Abstract | PDF (616 KB) 

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    Chapter 8: Parallel scaling

    pp. 199-216 (18 pages)

    https://doi.org/10.1137/1.9781611976311.ch8

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    Chapter 9: Finite element method I: Nonlinear optimization

    pp. 217-241 (25 pages)

    https://doi.org/10.1137/1.9781611976311.ch9

    Abstract | PDF (653 KB) 

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    Chapter 10: Finite element method II: Naive and unstructured

    pp. 243-277 (35 pages)

    https://doi.org/10.1137/1.9781611976311.ch10

    Abstract | PDF (934 KB) 

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    Chapter 11: Advection without, and then with, diffusion

    pp. 279-313 (35 pages)

    https://doi.org/10.1137/1.9781611976311.ch11

    Abstract | PDF (1152 KB) 

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    Chapter 12: Inequality constraints

    pp. 315-330 (16 pages)

    https://doi.org/10.1137/1.9781611976311.ch12

    Abstract | PDF (724 KB) 

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    Chapter 13: Finite element method III: Firedrake and DMPlex

    pp. 331-341 (11 pages)

    https://doi.org/10.1137/1.9781611976311.ch13

    Abstract | PDF (408 KB) 

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    Chapter 14: Stokes equations (with Firedrake)

    pp. 343-370 (28 pages)

    https://doi.org/10.1137/1.9781611976311.ch14

    Abstract | PDF (775 KB) 

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    Appendix: Some numerical facts of life

    pp. 371-372 (2 pages)

    https://doi.org/10.1137/1.9781611976311.appa

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    Back Matter

    pp. 373-391 (21 pages)

    https://doi.org/10.1137/1.9781611976311.bm

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    ###-Book Description End-###
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