Intro; Preface; Acknowledgements; Contents; About the Editors; 1 An Introduction to Recent Developments in Numerical Methods for Partial Differential Equations; References; 2 An Introduction to the Theory of M-Decompositions; 2.1 Introduction; 2.2 What Motivated the Appearance of the M-Decompositions?; 2.2.1 DG Methods; 2.2.2 HDG Methods; 2.2.3 Local Spaces or Stabilization Functions; 2.3 The M-Decompositions; 2.3.1 Definition; 2.3.2 The HDG-Projection; 2.3.3 Estimates of the Projection of the Errors; 2.3.4 Local Postprocessing; 2.3.5 Approximation Properties of the HDG-Projection

2.4 A Construction of M-Decompositions2.4.1 A Characterization of M-Decompositions; 2.4.2 The General Construction; 2.5 Examples; 2.5.1 An Illustration of the Construction; 2.5.2 Triangular and Quadrilateral Elements; 2.5.3 General Polygonal Elements; 2.6 Extensions; Appendix: Proof of the Characterization of M-Decompositions; References; 3 Mimetic Spectral Element Method for Anisotropic Diffusion; 3.1 Introduction; 3.1.1 Overview of Standard Discretizations; 3.1.2 Overview of Mimetic Discretizations; 3.1.3 Outline of Chapter; 3.2 Anisotropic Diffusion/Darcy Problem; 3.2.1 Gradient Relation

3.2.2 Divergence Relation3.2.3 Dual Grids; 3.3 Mimetic Spectral Element Method; 3.3.1 One Dimensional Spectral Basis Functions; 3.3.2 Two Dimensional Expansions; 3.3.2.1 Expanding p (Direct Formulation); 3.3.2.2 Expanding u and p (Mixed Formulation); 3.4 Transformation Rules; 3.5 Numerical Results; 3.5.1 Manufactured Solution; 3.5.2 The Sand-Shale System; 3.5.3 The Impermeable-Streak System; 3.6 Future Work; References; 4 An Introduction to Hybrid High-Order Methods; 4.1 Introduction; 4.2 Discrete Setting; 4.2.1 Polytopal Mesh; 4.2.2 Regular Mesh Sequences; 4.2.3 Local and Broken Spaces

4.2.4 Projectors on Local Polynomial Spaces4.2.4.1 L2-Orthogonal Projector; 4.2.4.2 Elliptic Projector; 4.2.4.3 Approximation Properties; 4.3 Basic Principles of Hybrid High-Order Methods; 4.3.1 Local Construction; 4.3.1.1 Computing the Local Elliptic Projection from L2-Projections; 4.3.1.2 Local Space of Degrees of Freedom; 4.3.1.3 Potential Reconstruction Operator; 4.3.1.4 Local Contribution; 4.3.1.5 Consistency Properties of the Stabilization for Smooth Functions; 4.3.2 Discrete Problem; 4.3.2.1 Global Spaces of Degrees of Freedom; 4.3.2.2 Global Bilinear Form

4.3.2.3 Discrete Problem and Well-Posedness4.3.2.4 Implementation; 4.3.2.5 Local Conservation and Flux Continuity; 4.3.3 A Priori Error Analysis; 4.3.3.1 Energy Error Estimate; 4.3.3.2 Convergence of the Jumps; 4.3.3.3 L2-Error Estimate; 4.3.4 A Posteriori Error Analysis; 4.3.4.1 Error Upper Bound; 4.3.4.2 Error Lower Bound; 4.3.5 Numerical Examples; 4.3.5.1 Two-Dimensional Test Case; 4.3.5.2 Three-Dimensional Test Case; 4.3.5.3 Three-Dimensional Case with Adaptive Mesh Refinement; 4.4 A Nonlinear Example: The p-Laplace Equation; 4.4.1 Discrete W1,p-Norms and Sobolev Embeddings