Satyendra K. Tomar

h - p Spectral Element Method for Elliptic Problems on Non-smooth Domains Using Parallel Computers

We propose a new h-p spectral element method to solve elliptic boundary value problems with mixed Neumann and Dirichlet boundary conditions on non-smooth domains. The method is shown to be exponentially accurate and asymptotically faster than the standard h-p finite element method. The spectral element functions are fully non-conforming for pure Dirichlet problems and conforming only at the vertices of the elements for mixed problems, and hence, the dimension of the resulting Schur complement matrix is quite small. The method is a least-squares collocation method and the resulting normal equations are solved using preconditioned conjugate gradient method with an almost optimal preconditioner. The algorithm is suitable for a distributed memory parallel computer. The numerical results of a number of model problems are presented, which confirm the theoretical estimates.

A multilevel method for discontinuous Galerkin approximation of three‐dimensional anisotropic elliptic problems

We construct optimal order multilevel preconditioners for interior-penalty discontinuous Galerkin (DG) finite element discretizations of three-dimensional (3D) anisotropic elliptic boundary-value problems. In this paper, we extend the analysis of our approach, introduced earlier for 2D problems (SIAM J. Sci. Comput., accepted), to cover 3D problems. A specific assembling process is proposed, which allows us to characterize the hierarchical splitting locally. This is also the key for a local analysis of the angle between the resulting subspaces. Applying the corresponding two-level basis transformation recursively, a sequence of algebraic problems is generated. These discrete problems can be associated with coarse versions of DG approximations (of the solution to the original variational problem) on a hierarchy of geometrically nested meshes. A new bound for the constant γ in the strengthened Cauchy–Bunyakowski–Schwarz inequality is derived. The presented numerical results support the theoretical analysis and demonstrate the potential of this approach. Copyright

Multilevel Preconditioning of Two-dimensional Elliptic Problems Discretized by a Class of Discontinuous Galerkin Methods

We present optimal-order preconditioners for certain discontinuous Galerkin (DG) finite element discretizations of elliptic boundary value problems. We consider two variants of hierarchical splittings where the underlying finite element meshes are geometrically nested. A specific assembling process is proposed which facilitates the local analysis of the angle between the related subspaces. By applying the corresponding two-level basis transformation recursively, a sequence of algebraic problems is generated that can be associated with a hierarchy of coarse versions of DG approximations of the original problem. New bounds for the constant

Algebraic multilevel preconditioning in isogeometric analysis: Construction and numerical studies

\gamma

A posteriori error estimates for approximations of evolutionary convection–diffusion problems

in the strengthened Cauchy-Bunyakowski-Schwarz inequality are derived. The presented numerical results support the theoretical analysis and demonstrate the potential of this approach.

A Multilevel Method for Discontinuous Galerkin Approximation of Three-dimensional Elliptic Problems

Abstract We present algebraic multilevel iteration (AMLI) methods for isogeometric discretization of scalar second order elliptic problems. The construction of coarse grid operators and hierarchical complementary operators are given. Moreover, for a uniform mesh on a unit interval, the explicit representation of B-spline basis functions for a fixed mesh size h is given for p = 2 , 3 , 4 and for C 0 - and C p - 1 -continuity. The presented methods show h - and (almost) p -independent convergence rates. Supporting numerical results for convergence factor and iterations count for AMLI cycles ( V -, linear W -, nonlinear W -) are provided. Numerical tests are performed, in two-dimensions on square domain and quarter annulus, and in three-dimensions on quarter thick ring.

IETI - Isogeometric Tearing and Interconnecting.

We derive computable upper bounds for the difference between an exact solution of the evolutionary convection-diffusion problem and an approximation of this solution. The estimates are obtained by certain transformations of the integral identity that defines the generalized solution. These estimates depend on neither special properties of the exact solution nor its approximation and involve only global constants coming from embedding inequalities. The estimates are first derived for functions in the corresponding energy space, and then possible extensions to classes of piecewise continuous approximations are discussed. Bibliography: 7 titles.

Multigrid methods for isogeometric discretization

We construct optimal order multilevel preconditioners for interiorpenalty discontinuous Galerkin (DG) finite element discretizations of 3D elliptic boundary-value problems. A specific assembling process is proposed which allows us to characterize the hierarchical splitting locally. This is also the key for a local analysis of the angle between the resulting subspaces. Applying the corresponding two-level basis transformation recursively, a sequence of algebraic problems is generated. These discrete problems can be associated with coarse versions of DG approximations (of the solution to the original variational problem) on a hierarchy of geometrically nested meshes. The presented numerical results demonstrate the potential of this approach.