A third-order mixed finite-element method for the numerical solution of the biharmonic problem (Q1334790)

The author introduces a rectangular mixed finite element and proves that the element is third-order convergent for Dirichlet problems for the biharmonic equation. Denote the set of polynomials in \(x\) and \(y\), the two coordinate directions, of order \(p\) by \(P_ p\). The finite element is based on rectangles \(T_ h\) with sides parallel to the \(x\) and \(y\) axes, with largest side of length \(h\) and with aspect ratios bounded above and below. Each edge of each rectangle has four nodes, with an additional node in the center, for a total thirteen nodes. The trial functions defined on the rectangles consist of scalar functions \(\phi\), \(v_{xx}\), \(v_{xy}\), and \(v_{yy}\) (representing the solution and its three second partial derivatives). On each rectangle \(T_ h\), the functions \(\phi\) are \(C^ 0\), satisfy the boundary condition \(\phi = 0\) on the boundary, and are in the space \(P_ 3 \oplus \{x^ 3 y, x^ 2y^ 2, xy^ 3\}\). The functions \(v_{xy}\) are in the space \(P_ 2\); the functions \(v_{xx}\) are in the space \(P_ 2 \oplus \{xy^ 2, x^ 3\}\) and are continuous across vertical element edges; and the functions \(v_{yy}\) are in the space \(P_ 2 \oplus \{x^ 2y, y^ 3\}\) and are continuous across horizontal element edges. If \(\psi\) is the solution of the weak form of the biharmonic equation \[ \Delta^ 2 \psi = f \text{ in }\Omega,\quad \psi = \partial \psi/ \partial n = 0 \text{ on } \partial \Omega, \] and \(u_{\alpha \beta} = \partial^ 2 \psi / \partial \alpha \partial \beta\) for \(\alpha, \beta = x, y\), and if the subscript \(h\) indicates finite element approximation, then the author proves that \[ \| u_{xx} - u_{xx_ h} \|_ 0 + \| u_{xy} - u_{xy_ h} \|_ 0 + \| u_{yy} - u_{yy_ h} \|_ 0 + \| \psi - \psi_ h \|_ 1 \leq Ch^ 3. \]