A new class of iterative methods for nonselfadjoint or indefinite problems
SIAM Journal on Numerical Analysis
Two-grid Discretization Techniques for Linear and Nonlinear PDEs
SIAM Journal on Numerical Analysis
Local and parallel finite element algorithms based on two-grid discretizations
Mathematics of Computation
A two-grid discretization scheme for eigenvalue problems
Mathematics of Computation
SIAM Journal on Numerical Analysis
A Two-Grid Discretization Scheme for Semilinear Elliptic Eigenvalue Problems
SIAM Journal on Scientific Computing
A Defect Correction Scheme for Finite Element Eigenvalues with Applications to Quantum Chemistry
SIAM Journal on Scientific Computing
A Two-Grid Method of a Mixed Stokes-Darcy Model for Coupling Fluid Flow with Porous Media Flow
SIAM Journal on Numerical Analysis
Three-Scale Finite Element Discretizations for Quantum Eigenvalue Problems
SIAM Journal on Numerical Analysis
Local and parallel finite element algorithms for the stokes problem
Numerische Mathematik
Nonconforming finite element approximations of the Steklov eigenvalue problem
Applied Numerical Mathematics
Numerical Solution to a Mixed Navier-Stokes/Darcy Model by the Two-Grid Approach
SIAM Journal on Numerical Analysis
The adaptive finite element method based on multi-scale discretizations for eigenvalue problems
Computers & Mathematics with Applications
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This paper discusses highly efficient discretization schemes for solving self-adjoint elliptic differential operator eigenvalue problems. Several new two-grid discretization schemes, including the conforming and nonconforming finite element schemes, are proposed by combining the finite element method with the shifted-inverse power method for matrix eigenvalue problems. With these schemes, the solution of an eigenvalue problem on a fine grid $\pi_{h}$ is reduced to the solution of an eigenvalue problem on a much coarser grid $\pi_{H}$ and the solution of a linear algebraic system on the fine grid $\pi_{h}$. Theoretical analysis shows that the schemes have a high efficiency. For instance, the resulting solution can maintain an asymptotically optimal accuracy by using the conforming linear element or the nonconforming Crouzeix-Raviart element by taking $H=O(\sqrt[4]{h})$. Numerical experiments are presented to support the theoretical analysis. In addition, this paper establishes multigrid discretization schemes and proves their efficiency.