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Partial differential equations and finite element method. (English) Zbl 1092.65080

In the book an introduction in the analysis and numerical solution of partial differential equations (PDEs) is presented. In Chapters 1-4 an introduction into the theory of PDEs is given and the basic concept of the finite element method is described. Especially, nodal and hierarchical approaches for the construction of higher-order elements are discussed. Chapter 5 is devoted to the numerical solution of systems of ordinary differential equations arising from the semidiscretization of time-dependent PDEs.
In Chapter 6 the numerical solution of fourth-order problems (beam and plate bending problems) is explained. The mathematical model is described, the weak formulation is given and the unique solvability is proved. Again lowest-order and higher-order approximations are used in the finite element discretization. In Chapter 7 the numerical solution of problems of electromagnetics is discussed. A detailed description of the mathematical models is given and the concept of modern edge elements is explained.
Appendix A contains the basics of functional analysis and linear algebra which are necessary for understanding the theory and numerical treatment of partial differential equations. In Appendix B some finite element software is described and numerical examples arising from practical problems are reported. The book is useful for all people who are working in disciplines of computational engineering.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74K20 Plates
78A25 Electromagnetic theory (general)
35Q60 PDEs in connection with optics and electromagnetic theory
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
78M10 Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory
74S05 Finite element methods applied to problems in solid mechanics
35K15 Initial value problems for second-order parabolic equations
35J25 Boundary value problems for second-order elliptic equations
35J40 Boundary value problems for higher-order elliptic equations
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