New schemes with fractal error compensation for PDE eigenvalue computations

Abstract

With an error compensation term in the fractal Rayleigh quotient of PDE eigen-problems, we propose a new scheme by perturbing the mass matrix M ^h to $\tilde M^h = M^h + Ch^{2m} K^h$, where K ^h is the corresponding stiff matrix of a 2m − 1 degree conforming finite element with mesh size h for a 2m-order self-adjoint PDE, and the constant C exists in the priority error estimation λ ^h _j − λ _j ∼ Ch ^2m λ ² _j . In particular, for Laplace eigenproblems over regular domains in uniform mesh, e.g., cube, equilateral triangle and regular hexagon, etc., we find the constant $C = \frac{{\left\| {I - h^{ - 1} M^h } \right\|}} {{2\left\| {hK^h } \right\|}}$ and show that in this case the computation accuracy can raise two orders, i.e., from λ ^h_j − λ _j = O(h ²) to O(h ⁴). Some numerical tests in 2-D and 3-D are given to verify the above arguments.