Low-Order Preconditioning of High-Order Triangular Finite Elements

Abstract

We propose a new formulation of a low-order elliptic preconditioner for high-order triangular elements. In the preconditioner, the nodes of the low-order finite element problem do not necessarily coincide with the high-order nodes. Instead, the two spaces are connected using least squares projection operators. The effectiveness of the preconditioner is demonstrated to be highly sensitive to the location and number of vertices used to construct the low-order finite element mesh on each high-order element. We treat the number of low-order vertices and their locations as optimizable quantities and chose them to minimize the condition number of the preconditioned stiffness matrix on the reference element. We present computational results that demonstrate that the condition number of the preconditioned high-order stiffness matrix on the reference element can be improved. The best performing preconditioners are formed with low-order finite element meshes that have more vertices than the high-order element has degrees of freedom, and have nodes grouped close to the element edges.