Introduction

This handbook describes both the classical finite element method and the isogeometric finite element method. The first scientific papers on the finite element method come from 1940 by Richard Courant (professor of mathematics, born in Lubliniec, Poland in 1888 in the then German territory, immigrated to the USA) and Alexander Hrennikoff (professor of civil engineering, born in Russia, emigrated to Canada), and Feng Kang in China in 1950 [1], [2], [3]. This method became popular in the years 1960-1970 thanks to the work of Olgierd Ziemkiewicz (civil engineering professor, Polish descent, lived in the UK)(civil engineering professor, Polish descent, lived in UK) [4]. In recent years, the isogeometric finite element method, promoted by the team of Prof. T. J. R. Hughes, has become popular. It uses basis functions from the B-spline family, characterized by a higher degree of continuity \( C^k \) [5]. Parallel to the methods of isogeometric analysis, adaptive methods are being developed using the classical finite element method with hierarchical basis functions. \( hp \) adaptive algorithms, allowing for exponential convergence of solution accuracy in relation to the size of the computational grid, are being developed by the group with prof. Leszek Demkowicz (Polish mathematician and professor of mechanics, working at the University of Texas in Austin) [6], [7]. There are also attempts to combine adaptive methods with isogeometric analysis by creating new families of polynomials, possible to define on adaptive grids, enabling mixing polynomials from the B-spline family to various degrees [8].

The classical method of finite elements on regular meshes is a special case of the isogeometric finite element method. The classical method of finite elements on regular meshes is a special case of the isogeometric finite element method. The only difference is that the basis functions used in the classical finite element method are polynomials of degree p of continuity \( C^{p-1} \) inside the elements, and on the border of the finite elements they are of class. \( C^0 \). Isogeometric method of finite elements generalizes base functions into polynomials of degree p which can be class \( C^k \) over the entire calculation area.
They can also be classes \( C^{p-1} \) only inside elements and classes \( C^0 \) at the interface of the elements. In particular, the B-spline functions used in the isogeometric method of finite elements are defined by the so-called knot vectors. The repetition of knots at the element boundary yields B-spline functions equivalent to classical Lagrange polynomials.

The isogeometric finite element method is usually used on grids that are the image of regular (square or cubic) element groups, while the classical finite element method can use square or triangular elements in 2D and cubic, tetrahedral, prisms and pyramids in 3D. There are, however, modern methods for defining B-spline functions on triangular and tetrahedral elements, and then this equivalence (the fact that isogeometric finite element method increases the continuity of basic functions) is preserved.

The classical finite element method approximates scalar fields and vectors appearing in piecewise continuous engineering calculations and the isogeometric finite element method in a globally continuous way. There are of course computational problems for which the isogeometric method gives a better approximation, and computational problems for which the classical finite element method gives better approximations. What elements, in my opinion, should the textbook on classical finite element method contain?

1. Introduction of the finite element definition and classical functions base polynomials such as Lagrange polynomials and hierarchical polynomials used, for example, in the adaptive finite element method. These definitions should include at least rectangular elements and triangular in two dimensions.
2. Introduction of algorithms for the generation of computational meshes, in particular made of triangles and tetrahedra, and algorithmic adaptation of computational grids.
3. Presentation of a number of simple problems in a strong and weak form.
4. Presentation of algorithms for the generation of systems of equations, left matrix sides (called ‘stiffness matrix’ for some problems) and the right-hand side vector.
5. Presentation of solver algorithms used for solving systems of equations.
6. Presentation of sample computational problems with results.
7. Presentation of stabilization methods for computationally difficult problems, for example, the Discountinuous Galerkin (DG) method or the Streamline Upwind Petrov-Galerkin (SUPG) method.

Ad.1) Formal definitions of the classical finite element method are described in the modules of the chapter "Mathematical foundations of the finite element method". The book also includes a number of informal definitions. In particular, triangular elements are described in the chapter "Non-regular meshes"; Lagrange polynomials on rectangular and cubic elements are defined by repeating nodes in the node vector defining B-spline basis functions (remembering that Lagrange polynomial functions are included in B-spline functions) in the chapter "Discretization with different basis functions".
Ad.2) The mesh generation algorithms are described in detail in the chapter "Computational grids".
Ad.3) Weak and strong forms do not depend on the way of discretization. In the chapter "Weak formulations for different problems and methods" a number of variational formulations are given, whether or not we use the classical or the isogeometric method. We have, in particular, heat transport equations, convection-diffusion equations, the Stokes problem, and the equations of the linear elasticity problem. These chapters also contain the digitization usually done with the help of the isogeometric finite element method; however, in the general part on strong and variational formulations, they are independent methods of discretization.
Ad.4) The problem of the generation of systems of equations resulting from the discretization of the classical finite element method is illustrated in the chapter "Heat transfer with classical finite element method" for a two-dimensional case. Relevant algorithms for the classical finite element methods can be found in the chapter "Mathematical foundations of the finite element method". There you can also find an example of the one-dimensional classical finite element method.
Ad.5) The algorithms of solvers are described in the chapter "Solvers of linear equations generated from finite element method", in the modules "Gaussian elimination algorithm", "Gaussian elimination with a pivoting algorithm", "LU factorization algorithm", "Frontal solver algorithm", "Multi-frontal solver algorithm", "Direction-splitting algorithm", "Pre-conditioner".
Ad.6) In the chapter "Weak formulations for different problems and methods" and in the module "Heat transfer with classical finite element method" there is an example of a weak and strong formulation for the two-dimensional problem of heat transport using the classical finite element method, and a number of algorithms concerning the generation of a system of equations. The section "Mathematical foundations of the finite element method" provides an example of one-dimensional classical finite element method, and a series of useful algorithms.
Ad.7) The DG stabilization method is described in the chapter "Stabilization methods" and in the module "Stabilization of the Stokes problem with the Discontinuous Galerkin method (DG)". In the module "Stabilization of the advection-diffusion equations with the Streamline Upwind Petrov-Galerkin (SUPG)", the SUPG method is described which works for both classical and isogeometric finite element methods.

What elements in my opinion should the textbook on isogemetric finite element method include?

1. Introduction of B-spline basis functions defined on a group (a patch) of items. These definitions should include the method of defining functions with a knots vector.
2. Introduction to algorithms for mapping geometric objects by element patches in CAD systems, element-to-object geometric patch mapping, and an element patch of adaptive algorithms.
3. Presentation of a number of simple problems in a strong and weak form.
4. Presentation of algorithms for generation of systems of equations, the left matrix side (called ‘stiffness matrix’ for certain problems) and the right-side vector.
5. Presentation of solver algorithms used for solving systems of equations.
6. Presentation of sample computational problems with results.
7. Presentation of stabilization methods for computationally difficult problems, e.g., residual minimization methods, the SUPG method.

Ad.1) This aspect is described in detail in the chapter "Discretization with different basis functions", the modules "Linear basis functions", "Higher-order Ck basis functions in 1D", "Refined isogeometric analysis in 1D", "Two-dimensional generalization of basis functions with tensor products", "Three-dimensional generalization of basis functions with tensor products".
Ad.2) The aspect of adaptive calculations in the isogeometric method is briefly described in the chapter "Computational grids" and the module "Isogeometric analysis on adaptive grids". The mapping aspect of CAD objects on groups of elements has been omitted from this book due to its breadth and belonging to related (but different) topics related to modeling geometry in CAD systems.
Ad.3) This issue is illustrated in the chapters "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" and "Interpretation of the solution".
Ad.4) These algorithms are described in the chapter "Solvers of linear equations generated from finite element method", the modules "Gaussian elimination algorithm", "Gaussian elimination with a pivoting algorithm", "LU factorization algorithm", "Frontal solver algorithm", "Multi-frontal solver algorithm", "Direction-splitting algorithm" and "Pre-conditioner". As such, all these algorithms are independent of whether we use the classical or isogeometric finite element method. Additionally, the modules "Iterative solver algorithm" and "Selection of the solver based on problem kind" refer to the isogeometric finite element method.
Ad.5) The solvers algorithms are described in the chapter "Solvers of linear equations generated from finite element method" in the modules "Gaussian elimination algorithm", "Gaussian elimination with a pivoting algorithm", "LU factorization algorithm", "Frontal solver algorithm", "Multi-frontal solver algorithm", "Direction-splitting algorithm" and "Pre-conditioner".
Ad.6) In the chapters "Exemplary problem of two-dimensional bitmap projection" and "Weak formulations for different problems and methods", there are extensive calculation examples for the isogeometric finite element method provided.
Ad.7) The stabilization method of the isogeometric finite element method has been described in the "Stabilization methods" section in the modules "Stabilization of the Stokes problem with the residual minimization method" and "Stabilization of the advection-diffusion equations with the Streamline Upwind Petrov-Galerkin (SUPG)".
Additionally, the manual includes a chapter describing the extension of the finite element method for non-stationary problems (modeling the state of systems changing over time) and the aforementioned chapter introducing the mathematical foundations of the finite element method.

Please send all comments and questions regarding the content of the book to the e-mail address maciej.paszynski o agh.edu.pl.