New schemes with fractal error compensation for PDE eigenvalue computations

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 _j ^h ? λ _j ～ Ch ^2m λ _j ² . 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|}}{{2left| {hK^h } right|}}) and show that in this case the computation accuracy can raise two orders, i.e., from λ _j ^h ? 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.