Intro
Contents
Part XII First-order PDEs
56 Friedrichs' systems
56.1 Basic ideas
56.1.1 The fields mathcalK and mathcalAk
56.1.2 Integration by parts
56.1.3 The model problem
56.2 Examples
56.2.1 Advection-reaction equation
56.2.2 Darcy's equations
56.2.3 Maxwell's equations
56.3 Weak formulation and well-posedness
56.3.1 Minimal domain, maximal domain, and graph space
56.3.2 The boundary operators N and M
56.3.3 Well-posedness
56.3.4 Examples
57 Residual-based stabilization
57.1 Model problem
57.2 Least-squares (LS) approximation

57.2.1 Weak problem
57.2.2 Finite element setting
57.2.3 Error analysis
57.3 Galerkin/least-squares (GaLS)
57.3.1 Local mesh-dependent weights
57.3.2 Discrete problem and error analysis
57.3.3 Scaling
57.3.4 Examples
57.4 Boundary penalty for Friedrichs' systems
57.4.1 Model problem
57.4.2 Boundary penalty method
57.4.3 GaLS stabilization with boundary penalty
58 Fluctuation-based stabilization (I)
58.1 Discrete setting
58.2 Stability analysis
58.3 Continuous interior penalty
58.3.1 Design of the CIP stabilization
58.3.2 Error analysis

58.4 Examples
59 Fluctuation-based stabilization (II)
59.1 Two-scale decomposition
59.2 Local projection stabilization
59.3 Subgrid viscosity
59.4 Error analysis
59.5 Examples
60 Discontinuous Galerkin
60.1 Discrete setting
60.2 Centered fluxes
60.2.1 Local and global formulation
60.2.2 Error analysis
60.2.3 Examples
60.3 Tightened stability by jump penalty
60.3.1 Local and global formulation
60.3.2 Error analysis
60.3.3 Examples
61 Advection-diffusion
61.1 Model problem
61.2 Discrete setting
61.3 Stability and error analysis

61.3.1 Stability and well-posedness
61.3.2 Consistency/boundedness
61.3.3 Error estimates
61.4 Divergence-free advection
62 Stokes equations: Residual-based stabilization
62.1 Model problem
62.2 Discrete setting for GaLS stabilization
62.3 Stability and well-posedness
62.4 Error analysis
63 Stokes equations: Other stabilizations
63.1 Continuous interior penalty
63.1.1 Discrete setting
63.1.2 Stability and well-posedness
63.1.3 Error analysis
63.2 Discontinuous Galerkin
63.2.1 Discrete setting
63.2.2 Stability and well-posedness

63.2.3 Error analysis
Part XIII Parabolic PDEs
64 Bochner integration
64.1 Bochner integral
64.1.1 Strong measurability and Bochner integrability
64.1.2 Main properties
64.2 Weak time derivative
64.2.1 Strong and weak time derivatives
64.2.2 Functional spaces with weak time derivative
65 Weak formulation and well-posedness
65.1 Weak formulation
65.1.1 Heuristic argument for the heat equation
65.1.2 Abstract parabolic problem
65.1.3 Weak formulation
65.1.4 Example: the heat equation
65.1.5 Ultraweak formulation
65.2 Well-posedness