Basic Theory of Standard Finite Element Method 1
1.1 The Basic Principles of Functional Analysis 1
1.1.1 Linear Operator and Linear Functional 1
1.1.2 Orthogonal Proj ection and Riesz Representation Theorem 4
1.1.3 Smooth Approximation and Fundamental Lemma of Calculus of Variation 5
1.1.4 Generalized Derivatives and Sobolev Spaces 6
1.1.5 Imbedding and Trace Theorems of Sobolev Spaces 9
1.1.6 Equivalent Module (Norm) Theorem 11
1.1.7 Green’s Formulas， Riesz-Thorin，s Theorem，Interpolation Inequality， and Closed Range Theorem 18
1.1.8 Fixed Point Theorems 20
1.2 Well-Posedness of Partial Differential Equations 21
1.2.1 The Classification for the Partial Differential Equations 21
1.2.1.1 Physical Classification for Partial Differential Equations 21
1.2.1.2 Mathematical Classification for Partial Differential Equations 21
1.2.1.3 The Second-Order Eq (1.2.2) Does Not Change Its Fomi under the Invertible Transformation 22
1.2.1.4 The Classification According to Characteristic Line 23
1.2.1.5 The Classification for the System of Partial Differential Equations 25
1.2.2 Lax-Milgram Theorem 26
1.2.3 Examples of Application for the Lax-Milgram Theorem 29
1.2.4 Differentiability (Regularity) of Generalized Solutions 41
1.3 Basic Theories of Function Interpolations 44
1.3.1 Finite Element and Related Properties 44
1.3.2 Properties of Finite Element Space and Inverse Estimation Theorem 46
1.3.3 Function Interpolation and Properties 55
1.3.4 The Interpolation Estimates in the Sobolev Spaces 58
1.4 Function Interpolations on Triangle Elements 63
1.4.1 Lagrange Linear Interpolation on the Triangle Elements 63
1.4.2 Lagrange’s Quadratic Interpolation on the Triangle Elements 66
1.4.3 Lagrange’s Cubic Interpolation on the Triangle Elements 68
1.4.4 Restricted Lagrange Cubic Interpolation 71
1.4.5 Cubic Hermite Interpolation on the Triangle Elements 74
1.4.5.1 Complete Cubic Hermite Interpolation on the Triangle Elements 74
1.4.5.2 Restricted Hermite Cubic Interpolation on the Triangle Elements 78
1.4.6 Quintic Hermite Interpolation on the Triangle Elements 80
1.4.6.1 Quintic Hermite Interpolation with 21 Degrees of Freedom 80
1.4.6.2 Quintic Hermite Interpolation with 18 Degrees of Freedom 81
1.4.7 Clough Interpolation on the Triangular Elements 89
1.4.8 Modified Clough Interpolation on the Triangular Elements 90
1.4.9 Morley，s Interpolation on the Triangle Elements 91
1.5 Function Interpolation on the Tetrahedral Element 92
1.5.1 Lagrange Linear Interpolation on the Tetrahedral Elements 92
1.5.2 Lagrange Quadratic Interpolation on the Tetrahedrons 96
1.5.3 Lagrange Cubic Interpolation on the Tetrahedral Elements 98
1.5.4 Hermite Cubic Interpolation with 20 Degrees of Freedom on the Tetrahedral Elements 100
1.5.5 Restricted Hermite Cubic Interpolation on the Tetrahedral Elements with 16 Degrees of Freedom 105
1.6 Interpolation of Functions on Rectangular Elements 110
1.6.1 Bilinear Lagrange Interpolation on the Rectangular Element 111
1.6.2 Biquadratic Lagrange Interpolation on the Rectangles 114
1.6.3 Incomplete Biquadratic Lagrange Interpolation on the Rectangular Elements 116
1.6.4 Complete Bicubic Hermite Interpolation on Rectangles 119
1.6.5 Incomplete Bicubic Hermite Interpolation on Rectangular 122
1.7 Function Interpolation on Arbitrary Quadrilaterals 126
1.7.1 Bilinear Interpolation on the Arbitrary Quadrilateral 132
1.7.2 Complete Biquadratic Interpolation on the Quadrilateral 133
1.7.3 Incomplete Biquadratic Interpolation on the Quadrilateral 135
1 _ 8 Function Interpolation on Hexahedron Elements 136
1.8.1 Interpolation Basis Functions on the Standard Cube 137
1.8.1.1 Trilinear Interpolation Basis Functions on the Standard Cube 137
1.8.1.2 The Basis Functions of Incomplete Triquadratic Interpolation on the Standard Cube 138
1.8.2 Function Interpolation on the Arbitrary Hexahedron 139
1.8.2.1 Trilinear Interpolation on the Arbitrary Hexahedral Elements 146
1.8.2.2 Incomplete Triquadratic Interpolation on the Arbitrary Hexahedral Elements 146
1.9 Convergence and Error Estimates of Finite Element Solutions 148
1.9.1 Projection Theorem and Galerkin Approximation 148
1.9.2 Finite Element Approximation for the First Homogeneous Boundary Value Problem of Poisson Equation 150
1.9.2.1 Error Estimates for the Finite Element Solution of First Homogeneous Boundary Value Problem of Poisson Equation 150
1.9.2.2 L2 Projection and Its Properties 153
1.9.2.3 L°° Estimate for the FE Solution of First Homogeneous Boundary Value Problem of Poisson Equation 157
1.9.3 Error Analysis of Finite Element Solution for the Biharmonic Equation 158
1.9.3.1 The Conforming FE Analysis of Biharmonic Equation 158
1.9.3.2 The Non-Conforming FE Analysis of Biharmonic Equation 162
2 Basic Theory of Mixed Finite Element Method 17
