Theory and Practice of Finite Elements
The origins of the finite element method can be traced back to the 1950s when engineers started to solve numerically structural mechanics problems in aeronautics. Since then, the field of applications has widened steadily and nowadays encompasses nonlinear solid mechanics, fluid/structure interactions, flows in industrial or geophysical settings, multicomponent reactive turbulent flows, mass transfer in porous media, viscoelastic flows in medical sciences, electromagnetism, wave scattering problems, and option pricing (to cite a few examples). Numerous commercial and academic codes based on the finite element method have been developed over the years. The method has been so successful to solve Partial Differential Equations (PDEs) that the term "Finite Element Method" nowadays refers not only to the mere interpolation technique it is, but also to a fuzzy set of PDEs and approximation techniques. The efficiency of the finite element method relies on two distinct ingredi ents: the interpolation capability of finite elements (referred to as the approx imability property in this book) and the ability of the user to approximate his model (mostly a set of PDEs) in a proper mathematical setting (thus guar anteeing continuity, stability, and consistency properties). Experience shows that failure to produce an approximate solution with an acceptable accuracy is almost invariably linked to departure from the mathematical foundations. Typical examples include non-physical oscillations, spurious modes, and lock ing effects. In most cases, a remedy can be designed if the mathematical framework is properly set up.
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advection affine algorithm approximate problem approximation space array Assume assumptions Banach space barycentric coordinates bijective bilinear form BNB Theorem boundary conditions bounded coercivity condition number Consider constant continuous embedding convergence Corollary defined Definition degrees of freedom Delaunay triangulation denote derivative dimension Dirichlet problem discrete problem domain edges eigenvalue equation error estimates Example Exercise finite element method Galerkin method geometric global shape functions Hence Hilbert space holds homogeneous Dirichlet hypotheses implies independent of h inf-sup condition integral interpolation operator introduce iterative Jn Jn L2-norm Lagrange finite element Lagrange polynomials Lax-Milgram Lemma Lemma Let Q linear system mapping nodes norm notation optimal Owing polynomials Proof Proposition Prove quadrature Remark result satisfies Seek Uh shape-regular family simplex solve sparse matrix stability stiffness matrix Stokes problem surjective symmetric technique triangle vector space velocity vertices weak formulation well-posed well-posedness yields zero
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