## Functional Analysis and Boundary-Value Problems: An Introductory Treatment |

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approximate solution arbitrary basis functions belongs bilinear form boundary conditions called Cauchy sequence Chapter complete Consider continuous function denoted derivative differential equations discussion distribution domain Q elliptic error estimate Exercise find u e finite element method finite-dimensional functional analysis functions defined Furthermore Galerkin method given Green's formula Green's theorem Hence Hilbert space Hm(Q HS(Q inner product space interpolation interval L2-norm Lebesgue integral Lemma limit points linear functional linear operator linear space linearly independent LP(Q maps master element matrix neighbourhood nodes normed space obtain Oden one-to-one open set order 2m orthogonal projection orthonormal basis orthonormal set piecewise polynomials projection operator Proof properties real numbers result satisfies Section shown in Figure Sobolev spaces space H spaces Cm(Q span subset subspace sup-norm Suppose unique solution V-elliptic VBVP vector zero