Essential boundary conditions and multi-point constraints in finite element analysis (Q5949106)

Elliptic problems with nonstandard boundary conditions are treated as variational problems with restrictions NEWLINE\[NEWLINE\text{ minimize } \tfrac 12 x^T Sx-f^T x\quad \text{subject to } Cx=g.NEWLINE\]NEWLINE The saddle point equations \(Sx+C^T \lambda =f, ~Cx=g\) and their solvability are discussed from the algebraic viewpoint, i.e., \(g\in \text{Range}(C)\), \(\text{Ker} (S) \cap \text{Ker} (C) =\{0\}\), and \(C\) is of full rank. Applications to classic situations are elucidated. There is no hint on the analytic behaviour of this finite element approximation, i.e., the coercivity on the kernel and Brezzi's condition are not mentioned.