## The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations |

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adjoint analysis approximate solution assume Babuska basis functions bilinear form boundary conditions boundary value problem bounded chapter coefficients compute consider constant corners corresponding cubic defined denote depends derivatives Dirichlet discrete E(ft eigenvalue problem elements of type elliptic operator error estimates example exists finite difference finite element method Galerkin method given H ft Hence Hilbert spaces holds HS(ft independent of h inequality interpolation introduce inverse assumption Lagrange Lemma linear functional mapping Math Mathematics matrix mesh nodal nodes norm numerical integration numerical solution obtain operator parabolic parameter partial differential equations patch test points polygon polynomials properties prove quadratic quadrature scheme rate of convergence resp satisfies singular Sobolev spaces solved spaces H spline subspaces t,k)-system theorem theory tion trial functions triangle unique variables variational principle vector vertex zero