\(-\Delta= -\text{grad div}+ \text{rot rot}\) for matrices, with application to the finite element solution of the Stokes problem (Q2716799)

The paper provides a detailed study of the algebraic Velte decomposition of \({\mathbb R}^n\) connected with a tripel of real symmetric matrices \(A>0\), \(B,C \geq 0\) with nontrivial nullspace. If \(A=B+C\) then ker \(B\) and ker \(C\) are orthogonal in the sense of the scalar product induced by \(A\). The orthogonal decomposition of \({\mathbb R}^n\) into three orthogonal subspaces ker \(B\), ker \(C\) and \(W\) is a finite-dimensional equivalent to the Velte decomposition of the Sobolev space \(H^1_0(\Omega)\) goverend by the identity given in the title of the paper. The author gives necessary and sufficient conditions that \(W\) is not empty by considering eigenvalue problems and different equations. NEWLINENEWLINENEWLINEThe theory, which is an algebraic equivalent of the classical vector analysis, is applied to the finite-element solution of the two-dimensional Stokes problem. It is shown that the discretization with Scott-Vogelius elements provides an algebraic Velte decomposition, leading to information about the spectrum of the Schur complement.