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Contents xv

7.3 Fully Discrete Schemes ... .. .. ... .. .. ... .. 311
7.4 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
7.5 The Maximum Principle for the
One-Step-Theta Method .. .. .. ... .. .. ... .. 323
7.6 Order of Convergence Estimates .. ... .. .. ... .. 330
8 Iterative Methods for Nonlinear Equations 342
8.1 Fixed-Point Iterations . ... .. .. ... .. .. ... .. 344
8.2 Newton’s Method and Its Variants ... .. .. ... .. 348
8.3 Semilinear Boundary Value Problems for Elliptic
and Parabolic Equations . . . . . . . . . . . . . . . . . . 360

9 Discretization Methods
for Convection-Dominated Problems 368
9.1 Standard Methods and
Convection-Dominated Problems . . . . . . . . . . . . . 368
9.2 The Streamline-Diffusion Method . . . . . . . . . . . . . 375
9.3 Finite Volume Methods ... .. .. ... .. .. ... .. 383
9.4 The Lagrange–Galerkin Method . . . . . . . . . . . . . . 387

A Appendices 390
A.1 Notation . ... .. .. ... .. .. ... .. .. ... .. 390
A.2 Basic Concepts of Analysis . . . . . . . . . . . . . . . . . 393
A.3 Basic Concepts of Linear Algebra . . . . . . . . . . . . . 394
A.4 Some Definitions and Arguments of Linear
Functional Analysis . . . . . . . . . . . . . . . . . . . . . 399
A.5 Function Spaces .. .. ... .. .. ... .. .. ... .. 404
References: Textbooks and Monographs 409

References: Journal Papers 412

Index 415