Local Jacobi Operators and Applications to the $p$-Version of Finite Element Method in Two Dimensions

• Authors: Benqi Guo;Ivo Babuška
• Venue: SIAM Journal on Numerical Analysis
• Year: 2010

Based on the Chebyshev projection-interpolation on each edge of elements and the Chebyshev projection on each element, we have designed the local Jacobi operators $\Pi_{\Omega_j}$ on the triangular or quadrilateral element $\Omega_j$, $1\leq j\leq J$ such that $\Pi_{\Omega_j}u$ is a polynomial of degree $p$ on $\Omega_j$ which interpolates $u$ at the vertices of $\Omega_j$, coincides with the Chebyshev projection-interpolation of $u$ on the edges of $\Omega_j$, and possesses the best approximation to the smooth and singular functions $u$. By a simple assembly of $\Pi_{\Omega_j}u$, $1\leq j\leq J$ we construct a piecewise and globally continuous polynomial of degree $p$ which has the best approximation error bound locally and globally for singular as well as smooth solutions on general quasi-uniform meshes and satisfies the homogeneous Dirichlet boundary conditions. An application of the local Jacobi operators to the $p$-version of the finite element method associated with general meshes composed of (curvilinear) triangular and quadrilateral elements for problems on polygonal domains leads to the optimal convergence, which has been an open problem for more than three decades.