Symmetrization for fractional Neumann problems

Neumann–neumann Methods for Vector Field Problems

In this paper, we study some Schwarz methods of Neumann–Neumann type for some vector field problems, discretized with the lowest order Raviart–Thomas and Nédélec finite elements. We consider a hybrid Schwarz preconditioner consisting of a coarse component, which involves the solution of the original problem on a coarse mesh, and local ones, which involve the solution of Neumann problems on the ...

Balancing Neumann-Neumann Methods for Elliptic Optimal Control Problems

We present Neumann-Neumann domain decomposition preconditioners for the solution of elliptic linear quadratic optimal control problems. The preconditioner is applied to the optimality system. A Schur complement formulation is derived that reformulates the original optimality system as a system in the state and adjoint variables restricted to the subdomain boundaries. The application of the Schu...

Conjugate Points Revisited and Neumann-Neumann Problems

The theory of conjugate points in the calculus of variations is reconsidered with a perspective emphasizing the connection to finite-dimensional optimization. The object of central importance is the spectrum of the second-variation operator, analogous to the eigenvalues of the Hessian matrix in finite dimensions. With a few basic properties of this spectrum, one can gain a new perspective on th...

Neumann-Series Solution of Fractional Differential Equation

Dirichlet-neumann and Neumann-neumann Waveform Relaxation Algorithms for Parabolic Problems

We present and analyze waveform relaxation variants of the Dirichlet-Neumann and NeumannNeumann methods for parabolic problems. These methods are based on a non-overlapping spatial domain decomposition, and each iteration involves subdomain solves with Dirichlet boundary conditions followed by subdomain solves with Neumann boundary conditions. However, unlike for elliptic problems, each subdoma...