Luca F. Pavarino

A Polylogarithmic Bound for an Iterative Substructuring Method for Spectral Elements in Three Dimensions

Iterative substructuring methods form an important family of domain decomposition algorithms for elliptic finite element problems. A

Overlapping Schwarz methods for mixed linear elasticity and Stokes problems

p

Iterative Substructuring Methods for Spectral Element Discretizations of Elliptic Systems. II: Mixed Methods for Linear Elasticity and Stokes Flow

-version finite element method based on continuous, piecewise

Iterative Substructuring Methods for Spectral Element Discretizations of Elliptic Systems I: Compressible Linear Elasticity

Q_p

Iterative Substructuring Methods for Spectral Elements: Problems in Three Dimensions Based on Numerical Quadrature

functions is considered for second-order elliptic problems in three dimensions; this special method can also be viewed as a conforming spectral element method. An iterative method is designed for which the condition number of the relevant operator grows only in proportion to

Preconditioned conjugate residual methods for mixed spectral discretizations of elasticity and Stokes problems

(1+log p)^2 .

Effects of mechanical feedback on the stability of cardiac scroll waves: A bidomain electro-mechanical simulation study

This bound is independent of jumps in the coefficient of the elliptic problem across the interfaces between the subregions. Numerical results are also reported which support the theory.

Monophasic action potentials generated by bidomain modeling as a tool for detecting cardiac repolarization times

Overlapping Schwarz preconditioners are introduced and studied for saddle point problems with a penalty term, such as Stokes equations and mixed formulations of linear elasticity. These preconditioners are based on the solution of local saddle point problems on overlapping subdomains and the solution of a coarse saddle point problem. Numerical experiments show that these are parallel and scalable preconditioners, since the rate of convergence of the preconditioned operator is independent of the mesh size h, the number of subdomains N, and the penalty parameter.

T wave polarity of simulated electrocardiograms: influence of transmural heterogeneity

Iterative substructuring methods are introduced and analyzed for saddle point problems with a penalty term. Two examples of saddle point problems are considered: The mixed formulation of the linear elasticity system and the generalized Stokes system in three dimensions. These problems are discretized with %mixed spectral element methods. The resulting stiffness matrices are symmetric and indefinite. The interior unknowns of each element are first implicitly eliminated by using exact local solvers. The resulting saddle point Schur complement is solved with a Krylov space method with block preconditioners. The velocity block can be approximated by a domain decomposition method, e.g., of wire basket type, which is constructed from a local solver for each face of the elements, and a coarse solver related to the wire basket of the elements. The condition number of the preconditioned operator is independent of the number of spectral elements and is bounded from above by the product of the square of the logarithm of the spectral degree and the inverse of the discrete inf-sup constant of the problem.

Domain decomposition algorithms for thep-version finite element method for elliptic problems

An iterative substructuring method for the system of linear elasticity in three dimensions is introduced and analyzed. The pure displacement formulation for compressible materials is discretized with the spectral element method. The resulting stiffness matrix is symmetric and positive definite. The proposed method provides a domain decomposition preconditioner constructed from local solvers for the interior of each element and for each face of the elements and a coarse, global solver related to the wire basket of the elements. As in the scalar case, the condition number of the preconditioned operator is independent of the number of spectral elements and grows as the square of the logarithm of the spectral degree.