Finite elements ; 2: Galerkin approximation elliptic and mixed PDEs Part V: Weak formulations and well posedness Weak formulation of model problems Main results on well posedness Part VI: Galerkin approximation Basic error analysis Error analysis with variational crimes Linear algebra Sparse matrices Quadratures Part VII: Elliptic PDEs: conforming approximation Scalar second order elliptic PDEs H1 conforming approximation (I) H1 conforming approximation (II) A posteriori error analysis The Helmholtz problem Part VIII: Elliptic PDEs: nonconforming approximation Crouzeix Raviart approximation Nitsches boundary penalty method Discontinuous Galerkin Hybrid high order methods Contrasted diffusivity (I) Contrasted diffusivity (II) Part IX: Vector valued elliptic PDEs Linear elasticity Maxwells equations: H(curl) approximation Maxwells equations: control on the divergence Maxwells equations: further topics Part X: Eigenvalue problems Symmetric elliptic eigenvalue problems Symmetric operators conforming approximation Nonsymmetric problems Part XI: PDEs in mixed form Well posedness for PDEs in mixed form Mixed finite element approximation Darcys equations Potential and flux recovery Stokes equations: Basic ideas Stokes equations: Stable Pairs (I) Stokes equations: Stable pairs (II) Appendices Bijective operators in Banach spaces. This book is the second volume of a three part textbook suitable for graduate coursework professional engineering and academic research. It is also appropriate for graduate flipped classes. Each volume is divided into short chapters. Each chapter can be covered in one teaching unit and includes exercises as well as solutions available from a dedicated website. The salient ideas can be addressed during lecture with the rest of the content assigned as reading material. To engage the reader the text combines examples basic ideas rigorous proofs and pointers to the literature to enhance scientific literacy. Volume II is divided into 32 chapters plus one appendix. The first part of the volume focuses on the approximation of elliptic and mixed PDEs beginning with fundamental results on well posed weak formulations and their approximation by the Galerkin method. The material covered includes key results such as the BNB theorem based on inf sup conditions Céas and Strangs lemmas and the duality argument by Aubin and Nitsche. Important implementation aspects regarding quadratures linear algebra and assembling are also covered. The remainder of Volume II focuses on PDEs where a coercivity property is available. It investigates conforming and nonconforming approximation techniques (Galerkin boundary penalty Crouzeix—Raviart discontinuous Galerkin hybrid high order methods). These techniques are applied to elliptic PDEs (diffusion elasticity the Helmholtz problem Maxwells equations) eigenvalue problems for elliptic PDEs and PDEs in mixed form (Darcy and Stokes flows). Finally the appendix addresses fundamental results on the surjectivity bijectivity and coercivity of linear operators in Banach spaces.