目錄
1 Basic Theory of Standard Finite Element Method
1.1 The Basic Principles of Functional Analysis
1.1.1 Linear Operator and Linear Functional
1.1.2 Orthogonal Projection and Riesz Representation Theorem
1.1.3 Smooth Approximation and Fundamental Lemma of Calculus of Variation
1.1.4 Generalized Derivatives and Sobolev Spaces
1.1.5 Imbedding and Trace Theorems of Sobolev Spaces
1.1.6 Equivalent Module (Norm) Theorem
1.1.7 Green's Formulas, Riesz-Thorin's Theorem, Interpolation Inequality, and Closed Range Theorem
1.1.8 Fixed Point Theorems
1.2 Well-Posedness of Partial Differential Equations
1.2.1 The Classification for the Partial Differential Equations
1.2.1.1 Physical Classification for Partial Differential Equations
1.2.1.2 Mathematical Classification for Partial Differential Equations
1.2.1.3 The Second-Order Eq. (1.2.2) Does Not Change Its Form under the Invertible Transformation
1.2.1.4 The Classification According to Characteristic Line
1.2.1.5 The Classification for the System of Partial Differential Equations
1.2.2 Lax-Milgram Theorem
1.2.3 Examples of Application for the Lax-Milgram Theorem
1.2.4 Differentiability (Regularity) of Generalized Solutions
1.3 Basic Theories of Function Interpolations
1.3.1 Finite Element and Related Properties
1.3.2 Properties of Finite Element Space and Inverse Estimation Theorem
1.3.3 Function Interpolation and Properties
1.3.4 The Interpolation Estimates in the Sobolev Spaces
1.4 Function Interpolations on Triangle Elements
1.4.1 Lagrange Linear Interpolation on the Triangle Elements
1.4.2 Lagrange's Quadratic Interpolation on the Triangle Elements
1.4.3 Lagrange's Cubic Interpolation on the Triangle Elements
1.4.4 Restricted Lagrange Cubic Interpolation
1.4.5 Cubic Hermite Interpolation on the Triangle Elements
1.4.5.1 Complete Cubic Hermite Interpolation on the Triangle Elements
1.4.5.2 Restricted Hermite Cubic Interpolation on the Triangle Elements
1.4.6 Quintic Hermite Interpolation on the Triangle Elements
1.4.6.1 Quintic Hermite Interpolation with 21 Degrees of Freedom
1.4.6.2 Quintic Hermite Interpolation with 18 Degrees of Freedom
1.4.7 Clough Interpolation on the Triangular Elements
1.4.8 Modified Clough Interpolation on the Triangular Elements
1.4.9 Morley's Interpolation on the Triangle Elements
1.5 Function Interpolation on the Tetrahedral Element
1.5.1 Lagrange Linear Interpolation on the Tetrahedral Elements
1.6.5 Incomplete Bicubic Hermite Interpolation on Rectangular...
1.7 Function Interpolation on Arbitrary Quadrilaterals
1.7.1 Bilinear Interpolation on the Arbitrary Quadrilateral
2 Basic Theory of Mixed Finite Element Method
3 Mixed Finite Element Methods for the Unsteady Partial Differential Equations
4 The Reduced Dimension Methods of Finite Element Subspaces for Unsteady Partial Differential Equations
5 The Reduced Dimension of Finite Element Solution
Coefficient Vectors for Unsteady Partial Differential Equations
Postscript and Author's Own Statement
Bibliography
Index
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