A Rational Finite Element Basis

Name: A Rational Finite Element Basis
Brand: Academic Press
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Academic Press

Erschienen am 14. Mai 2014

347 Seiten

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Front Cover
A Rational Finite Element Basis
Copyright Page
CONTENTS
PREFACE
THEOREMS AND LEMMAS
Chapter 1. PATCHWORK APPROXIMATION IN NUMERICAL ANALYSIS
1.1 Wedges and Pyramids
1.2 Definitions and Notation
1.3 Continuity
1.4 Patchwork Approximation Spaces and Convergence
1.5 Wedge Properties
1.6 Isoparametric Coordinates
1.7 Generalizations to Sides of Higher Order and to Three-Dimensional Elements
1.8 Remarks and References
Chapter 2. THE QUADRILATERAL
2.1 Inadequacy of Polynomials
2.2 Rational Wedges
2.3 Areal Coordinates as Limits of Rational Wedges
2.4 An Example of Quadrilateral Wedges
2.5 Projective Coordinates
2.6 Polygons?
Chapter 3. RATIONAL WEDGES FOR SELECTED POLYCONS
3.1 The 3-Con of Order Four
3.2 The 4-Con of Order Five
3.3 The Pentagon
3.4 Some Elementary Congruences
3.5 Wedges for 3-Cons of Orders Five and Six
3.6 Two-sided Elements
3.7 Related Studies
Chapter 4. ALGEBRAIC GEOMETRY FOUNDATIONS
4.1 Motivation
4.2 Homogeneous Coordinates and the Projective Plane
4.3 Intersection of Plane Curves
4.4 The Fundamental Congruence Theorem
4.5 Associated Points
4.6 Resolution of Singularities
4.7 Remarks and References
Chapter 5. RATIONAL WEDGE CONSTRUCTION FOR POLYCONS AND POLYPOLS
5.1 Polycon Wedge Construction
5.2 Verification of Polycon Wedge Properties
5.3 The Case of the Vanishing Denominator
5.4 Polypols and Deficit Intersection Points
5.5 Polypol Wedge Numerators and Adjunct Intersection Points
5.6 Illustrative Polycubes
Chapter 6. APPROXIMATION OF HIGHER DEGREE
6.1 Data Fitting
6.2 Degree Two Approximation
6.3 Degree Three and Higher Degree Approximation
6.4 Intermediate Approximation
6.5 Higher Degree Approximation on Polypols
6.6 A Concise Algebraic Geometry Analysis
6.7 Algebrais Reticulation
Chapter 7. THREE-DIMENSIONAL APPROXIMATION
7.1 Definitions and Background
7.2 Triangular Prisms and Hexahedra
7.3 Polyhedra
7.4 Polycondra
7.5 The Adjoint of a Well-Set Polypoldron
7.6 Polypoldra Nodes and Adjacent Factors for Degree k Approximation
7.7 Attainment of Degree k Approximation
Chapter 8. A RATIONAL SOLUTION TO AN IRRATIONAL PROBLEM
8.1 Irrational Wedges
8.2 The Method of Descent
8.3 Wedges for an Ill-Set Polycon
8.4 Nonconvex Quadrilaterals
8.5 Remarks
Chapter 9. FINITE ELEMENT DISCRETIZATION
9.1 Introductory Remarks
9.2 Some Simple Quadrature Formulas
9.3 Consistent Quadrature and the Patch Test
9.4 Triangle Averaging
9.5 Mosaic Discretization
9.6 A Discrete Laplacian for Quadrilaterals
9.7 Harmonious Discretization
Chapter 10. TWO-LEVEL COMPUTATION
10.1 Recapitulation
10.2 Synthesis
10.3 Coarse Mesh Rebalancing
10.4 Concluding Remarks
REFERENCES
INDEX

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