• Preface: Part I (12)
• Preface: Parts II and III (14)
  • Part II: Advances in Theory (15)
• Acknowledgements (16)
• Contents (18)
• Theorems and Lemmas (22)
• Part I A Rational Finite Element Basis (23)
  • 1 Patchwork Approximation in Numerical Analysis (24)
    • 1.1 Wedges and Pyramids (24)
    • 1.2 Definitions and Notation (27)
    • 1.3 Continuity (32)
    • 1.4 Patchwork Approximation Spaces and Convergence (34)
    • 1.5 Wedge Properties (37)
    • 1.6 Isoparametric Coordinates (38)
    • 1.7 Generalizations to Sides of Higher Order and to Three-Dimensional Elements (41)
    • 1.8 Remarks and References (42)
  • 2 The Quadrilateral (43)
    • 2.1 Inadequacy of Polynomials (43)
    • 2.2 Rational Wedges (43)
    • 2.3 Areal Coordinates as Limits of Rational Wedges (47)
    • 2.4 An Example of Quadrilateral Wedges (48)
    • 2.5 Projective Coordinates (49)
    • 2.6 Polygons? (54)
  • 3 Rational Wedges for Selected Polycons (57)
    • 3.1 The 3-Con of Order Four (57)
    • 3.2 The 4-Con of Order Five (64)
    • 3.3 The Pentagon (68)
    • 3.4 Some Elementary Congruences (68)
    • 3.5 Wedges for 3-Cons of Orders Five and Six (72)
    • 3.6 Two-Sided Elements (77)
    • 3.7 Related Studies (79)
  • 4 Algebraic Geometry Foundations (81)
    • 4.1 Motivation (81)
    • 4.2 Homogeneous Coordinates and the Projective Plane (82)
    • 4.3 Intersection of Plane Curves (83)
    • 4.4 The Fundamental Congruence Theorem (88)
    • 4.5 Associated Points (93)
    • 4.6 Resolution of Singularities (95)
    • 4.7 Remarks and References (102)
  • 5 Rational Wedge Construction for Polycons and Polypols (103)
    • 5.1 Polycon Wedge Construction (103)
    • 5.2 Verification of Polycon Wedge Properties (113)
    • 5.3 The Case of the Vanishing Denominator (116)
    • 5.4 Polypols and Deficit Intersection Points (125)
    • 5.5 Polypol Wedge Numerators and Adjunct Intersection Points (128)
    • 5.6 Illustrative Polycubes (130)
  • 6 Approximation of Higher Degree (134)
    • 6.1 Data Fitting (134)
    • 6.2 Degree Two Approximation (135)
    • 6.3 Degree Three and Higher Degree Approximation (141)
    • 6.4 Intermediate Approximation (144)
    • 6.5 Higher Degree Approximation on Polypols (146)
    • 6.6 A Concise Algebraic Geometry Analysis (147)
    • 6.7 Algebraic Reticulation (150)
  • 7 Three-Dimensional Approximation (152)
    • 7.1 Definitions and Background (152)
    • 7.2 Triangular Prisms and Hexahedra (155)
    • 7.3 Polyhedra (161)
    • 7.4 Polycondra (163)
    • 7.5 The Adjoint of a Well-Set Polypoldron (169)
      • 7.5.1 Conditions on Qm-4 (169)
      • 7.5.2 Existence of a Unique Adjoint (171)
      • 7.5.3 Wedge Regularity (173)
    • 7.6 Polypoldra Nodes and Adjacent Factors for Degree k Approximation (173)
      • 7.6.1 Node Placement (173)
      • 7.6.2 Adjacent Factors at Interior Nodes (174)
      • 7.6.3 Adjacent Factors for Face Nodes (174)
      • 7.6.4 Adjacent Factors for Edge Nodes (174)
      • 7.6.5 Adjacent Factors for Vertex Nodes (175)
    • 7.7 Attainment of Degree k Approximation (176)
  • 8 A Rational Solution to an Irrational Problem (177)
    • 8.1 Irrational Wedges (177)
    • 8.2 The Method of Descent (179)
    • 8.3 Wedges for an Ill-Set Polycon (182)
    • 8.4 Nonconvex Quadrilaterals (184)
    • 8.5 Remarks (187)
  • 9 Finite Element Discretization (189)
    • 9.1 Introductory Remarks (189)
    • 9.2 Some Simple Quadrature Formulas (190)
    • 9.3 Consistent Quadrature and the Patch Test (196)
    • 9.4 Triangle Averaging (201)
    • 9.5 Mosaic Discretization (203)
    • 9.6 A Discrete Laplacian for Quadrilaterals (207)
    • 9.7 Harmonious Discretization (212)
  • 10 Two-Level Computation (224)
    • 10.1 Recapitulation (224)
    • 10.2 Synthesis (225)
    • 10.3 Coarse Mesh Rebalancing (227)
    • 10.4 Concluding Remarks (227)
• Part II Advances in Theory (229)
  • 11 Two Dimensions (230)
    • 11.1 Convex Polygons (230)
    • 11.2 Polycons (234)
    • 11.3 Well-Set Concave Polycons (239)
    • 11.4 Positivity (240)
    • 11.5 Polypols: Elements with Sides of Order Greater Than Two (241)
    • 11.6 Integral Based Barycentric Coordinates (243)
    • 11.7 A Review of Region Partitioning (244)
    • 11.8 Stencils (244)
    • 11.9 Mapping (247)
  • 12 Higher Dimensions (249)
    • 12.1 GADJ for Convex Polyhedra (249)
    • 12.2 Polypoldra (251)
    • 12.3 Polytopes (252)
    • 12.4 A 4D Example (253)
  • 13 Forty Years After (255)
    • 13.1 Construction (255)
    • 13.2 Accuracy (255)
• Part III Computer Programs (257)
  • 14 Computer Programs (258)
• Relevant Work (293)
  • Search Words (293)
• References (294)
• Index (297)