Matrix-free methods for local elliptic subproblems in fictitious space preconditioning

Develop efficient matrix-free algorithms for solving the decoupled local elliptic Dirichlet problems on the interior of each triangular element that arise in the fictitious space preconditioner defined by element-wise elliptic projection from the Duffy-transformed high-order H1 space to the standard P_N Lagrange space, so as to make the preconditioner practical and scalable at high polynomial degrees.

Background

In the fictitious space framework, the standard P_N Lagrange finite element space is preconditioned using an auxiliary space consisting of the Duffy-based high-order H1 space on triangles. The transfer operator R is defined by local elliptic projection: on each triangle, the projected function matches the auxiliary function on the boundary and solves a homogeneous Laplace equation in the interior.

Implementing this preconditioner requires solving independent elliptic problems on the interior of each triangle. These element-local subproblems induce block-diagonal structures whose block sizes grow quadratically with the polynomial degree, making their efficient treatment crucial for high-order performance. The paper highlights the need for matrix-free approaches to avoid assembling dense local matrices and to maintain overall scalability.