Computational Framework for the Finite Element Method in MATLAB® and Python

1. Finite Element Method for the One-Dimensional Boundary Value Problem. 1.1 Formulation of the Problem. 1.2 Integral Equation. 1.3 Lagrange Interpolating Polynomials. 1.4 Illustrative Problem. 1.5 Algorithms of The Finite Element Method. 1.6. Quadrature Rules. 1.7. Defining Parameters of the FEM. 2. Programming One-Dimensional Finite Element Method. 2.1 Sparse Matrices in MATLAB. 2.2 Input Data Structures. 2.3 Coding Quadrature Rules. 2.4 Interpolating and Differentiating Matrices. 2.5. Calculating and Assembling Fem Matrices. 2.6 Python Implementation. 3. Finite Element Method for the Two-Dimensional Boundary Value Problem. 3.1 Model Problem. 3.2 Finite Elements Definition. 3.3. Triangulation Examples. 3.4. Linear System of the FEM. 3.5 Stiffness Matrix and Forcing Vector. 3.6. Algorithm of Solving Problem. 4. Building Two-Dimensional Meshes. 4.1. Defining Geometry. 4.2. Representing Meshes in Matrix Form for Linear Interpolation Functions. 4.3. Complementary Mesh. 4.4. Building Meshes in MATLAB. 4.5. Building Meshes in Python. 5. Programming Two-Dimensional Finite Element Method. 5.1. Assembling Global Stiffness Matrix. 5.2. Assembling Global Forcing Vector. 5.3. Calculating Local Stiffness Matrices. 5.4. Calculating Equation Coefficients. 5.5. Calculating Global Matrices. 5.6. Calculating Boundary Conditions. 5.7. Assembling Boundary Conditions 5.8. Solving Example Problem. 6. Nonlinear Basis Functions. 6.1. Linear Triangular Elements. 6.2. Curvilinear Triangular Elements. 6.3. Stiffness Matrix with Quadratic Basis. Conclusion Appendix A. Variational Formulation of a BVP. Appendix B. Discussion of Global Interpolation. Appendix C. Interpolatory Quadrature Formulas. Appendix D. Quadrature Rules and Orthogonal Polynomials. Appendix E. Computational Framework in Python.