In this talk we present a multiscale adaptive finite element method for PDE eigenvalue problems which will use one scale, e.g., P1 finite elements, to approximate the solution and finer scale, e.g., P2 finite elements, to capture the approximate residual. Starting from the results of Grubišić and Ovall 2009 on the reliable and efficient asymptotically exact a posteriori hierarchical error estimators in the selfadjoint case, we explore the possibility to use the enhanced Ritz values and vectors to restart the iterative algebraic procedures within the adaptive algorithm. Using higher order hierarchical polynomial finite element basis, as indicated by Bank 1996 and by Le Borne and Ovall 2012, our method generates discretization matrices which are almost diagonal. This construction can be repeated for the complements of higher (even) order polynomials and yields a structure which is particularly suitable for designing computational algorithms with low complexity. We present some preliminary numerical results for the symmetric eigenvalue problems.
This is a joint work with L. Grubišić (University of Zagreb) and J. Ovall (Portland State University).