Classical and isogeometric finite element method

Preface Introduction to the finite element method handbook 1. Basics of finite element method Exemplary problem of two-dimensional bitmap projection Approximation with B-spline basis functions Derivation of the system of linear equations Generation of the system of linear equations with the analytical method The solution of the system of linear equations Interpretation of the solution MATLAB implementation of the bitmap projection problem 2. Discretization with different basis functions Linear basis functions Higher-order Ck basis functions in 1D Lagrange polynomials Refined isogeometric analysis in 1D Two-dimensional generalization of basis functions with tensor products Three-dimensional generalization of basis functions with tensor products Non-regular meshes MATLAB implementation of the generation of basis functions based on the knot vector 3. Weak formulations for different problems and methods Heat transfer with isogeometric finite element method Heat transfer with classical finite element method Advection-diffusion equations Stokes problem Linear elasticity Weak formulations and numerical integration MATLAB implementation of the heat transfer problem MATLAB implementation of the advection-diffusion with Galerkin method 4. Solvers of linear equations generated from finite element method Gaussian elimination algorithm Gaussian elimination with a pivoting algorithm LU factorization algorithm Algorithm of LU factorization with pivoting Frontal solver algorithm Multi-frontal solver algorithm Direction-splitting algorithm Pre-conditioner Iterative solver algorithm Selection of the solver based on problem kind MATLAB implementation of the alternating-direction solver for the bitmap projection problem 5. Stabilization methods Stabilization of finite element method Stabilization of the advection-diffusion equations with the residual minimization method Stabilization of the advection-diffusion equations with the Streamline Upwind Petrov-Galerkin (SUPG) Stabilization of the Stokes problem with the residual minimization method Stabilization of the Stokes problem with the Discontinuous Galerkin method (DG) MATLAB implementation of the advection-diffusion problem with SUPG MATLAB implementation of the advection-diffusion problem with residual minimization method 6. Computational grids Generation and adaptation of computational grids Algorithm of adaptation of triangular and tetrahedral elements Algorithm of h-adaptation Algorithm of p-adaptation Algorithm of hp-adaptation Approximation on the mesh with hanging nodes Isogeometric analysis on adaptive grids MATLAB implementation of the adaptive algorithm for the bitmap projection problem 7. Finite element method for non-stationary and non-linear problems Non-stationary problems Non-stationary problems as a generalization of a sequence of the isogeometric L2 projections Examples of differential operators and right-hand-sides for selected problems Explicit method Implicit method Crank-Nicolson scheme Alpha scheme Example of the non-stationary linear problem: propagation of pollution MATLAB implementation of the alpha scheme for the heat transfer problem Example of the non-stationary non-linear problem: non-linear flow in heterogeneous media 8. Mathematical foundations of the finite element method Functional spaces One-dimensional finite element The one-dimensional strong and weak formulation for the elliptic problem One-dimensional finite element method Example of a one-dimensional finite element method Two-dimensional finite element The two-dimensional strong and weak formulation for the elliptic problem Two-dimensional finite element method Adaptive algorithm The convergence of the finite element method Isogeometric finite element method The future of the finite element method Appendices Appendix 1 Appendix 2 Appendix 3 Appendix 3A Appendix 4 Appendix 4A Appendix 5 Appendix 5A Appendix 6 Appendix 7