## Analysis of a Finite Element Method: PDE/PROTRANThis text can be used for two quite different purposes. It can be used as a reference book for the PDElPROTRAN user· who wishes to know more about the methods employed by PDE/PROTRAN Edition 1 (or its predecessor, TWODEPEP) in solving two-dimensional partial differential equations. However, because PDE/PROTRAN solves such a wide class of problems, an outline of the algorithms contained in PDElPROTRAN is also quite suitable as a text for an introductory graduate level finite element course. Algorithms which solve elliptic, parabolic, hyperbolic, and eigenvalue partial differential equation problems are pre sented, as are techniques appropriate for treatment of singularities, curved boundaries, nonsymmetric and nonlinear problems, and systems of PDEs. Direct and iterative linear equation solvers are studied. Although the text emphasizes those algorithms which are actually implemented in PDEI PROTRAN, and does not discuss in detail one- and three-dimensional problems, or collocation and least squares finite element methods, for example, many of the most commonly used techniques are studied in detail. Algorithms applicable to general problems are naturally emphasized, and not special purpose algorithms which may be more efficient for specialized problems, such as Laplace's equation. It can be argued, however, that the student will better understand the finite element method after seeing the details of one successful implementation than after seeing a broad overview of the many types of elements, linear equation solvers, and other options in existence. |

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### Sommario

1 | |

Elliptic ProblemsForming the Algebraic Equations | 22 |

Elliptic ProblemsSolving the Algebraic Equations | 50 |

Parabolic Problems | 77 |

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algorithm approximating assumed backward difference method band solver basis functions boundary conditions BOUNDARY FORCE calculated components conjugate gradient conjugate gradient method convergence Crank-Nicolson Crank-Nicolson method cubic curved D3EST Default defined in GLOBAL derivatives diagonal dimensional discretization discretization error DISPLACEMENTS DOUBLE PRECISION Structure ds R 3R2 dxdy eigenfunction eigenvalue problem error evaluated Expressions involving constants Figure filename finite difference method finite element formula FORTRAN frontal method frontal solver Galerkin given grid hyperbolic problems iarcl initial triangulation INTEGER Structure integration inverse power method isoparametric Jacobian matrix keyword Lanczos Iteration Lanczos method linear system Newton iteration Newton's method nonlinear nonzero normalized triangle NOUPDATE nset O(dt output PDE/PROTRAN PDE2D piecewise polynomial plot positive definite quadratic quartic elements REAL or DOUBLE satisfies Section shown solution solve specified step symmetric UPRINT USOL values variables defined vector velocity vertical WSOL XGRID zero