Piecewise solenoidal vector fields and the Stokes problem
SIAM Journal on Numerical Analysis
On the abstract theory of additive and multiplicative Schwarz algorithms
Numerische Mathematik
Two-level additive Schwarz preconditioners for nonconforming finite element methods
Mathematics of Computation
Domain decomposition: parallel multilevel methods for elliptic partial differential equations
Domain decomposition: parallel multilevel methods for elliptic partial differential equations
A Nonconforming Finite Element Method for the Stationary Navier--Stokes Equations
SIAM Journal on Numerical Analysis
Finite Element Method for Elliptic Problems
Finite Element Method for Elliptic Problems
Discontinuous hp-Finite Element Methods for Advection-Diffusion-Reaction Problems
SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems
SIAM Journal on Numerical Analysis
Poincaré-Friedrichs Inequalities for Piecewise H1 Functions
SIAM Journal on Numerical Analysis
Some nonstandard error analysis of discontinuous Galerkin methods for elliptic problems
Calcolo: a quarterly on numerical analysis and theory of computation
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We present some two-level non-overlapping additive and multiplicative Schwarz methods for a discontinuous Galerkin method for solving the biharmonic equation. We show that the condition numbers of the preconditioned systems are of the order O(H3/h3) for the non-overlapping Schwarz methods, where h and H stand for the fine mesh size and the coarse mesh size, respectively. The analysis requires establishing an interpolation result for Sobolev norms and Poincaré Friedrichs type inequalities for totally discontinuous piecewise polynomial functions. It also requires showing some approximation properties of the multilevel hierarchy of discontinuous Galerkin finite element spaces.