Fast hp preconditioners for elliptic and mixed problems

Many applications from science and engineering are mathematically described by partial differential equations. The finite element method (FEM) is certainly one of the most powerful tools for the computer simulation of such models. The p-version of the FEM operates on a fixed mesh, and increases the polynomial degree p per element. The advantage of this method is that smooth functions can be approximated very well by high order polynomials. Thus, the p-, and the hp- version of the FEM have become very popular discretization methods in mathematics and engineering. The discretization of a boundary value problem using FEM leads to a linear system of algebraic equations Ax=b. It is known from the literature that preconditioned Krylov subspace methods are among of the most efficient iterative solution methods for Ax=b. The convergence speed of this methods depends strongly on the choice of the considered preconditioners. In this project, several three dimensional boundary value problems will be discretized by the p- or the hp-version of the FEM using hexahedral elements. The corresponding linear system Ax=b will be solved by a preconditioned Krylov subspace method. We will develop several domain decomposition preconditioners for the system Ax=b such that the total solver time of Ax=b is proportionally to the dimension of the matrix A, i.e. the number of unknowns. We intend to use the tensor product structure of the hexahedral elements for the development of the preconditioners. We will investigate preconditioners for hp-FEM discretizations of scalar elliptic problems as well as for hp-FEM discretizations of the Lame equations of linear elasticity. Moreover, we will consider the Stokes problem as an example of a mixed problem. All preconditioners will be investigated theoretically and numerically in several numerical experiments.

Many applications from science and engineering are mathematically described by partial differential equations. The finite element method (FEM) is certainly one of the most powerful tools for the computer simulation of such models. The p-version of the FEM operates on a fixed mesh, and increases the polynomial degree p per element. The advantage of this method is that smooth functions can be approximated very well by high order polynomials. Thus, the p-, and the hp- version of the FEM have become very popular discretization methods in mathematics and engineering. The discretization of a boundary value problem using FEM leads to a linear system of algebraic equations Ax=b. It is known from the literature that preconditioned Krylov subspace methods are among of the most efficient iterative solution methods for Ax=b. The convergence speed of this methods depends strongly on the choice of the considered preconditioners. In this project, several three dimensional boundary value problems will be discretized by the p- or the hp-version of the FEM using hexahedral elements. The corresponding linear system Ax=b will be solved by a preconditioned Krylov subspace method. We will develop several domain decomposition preconditioners for the system Ax=b such that the total solver time of Ax=b is proportionally to the dimension of the matrix A, i.e. the number of unknowns. We intend to use the tensor product structure of the hexahedral elements for the development of the preconditioners. We will investigate preconditioners for hp-FEM discretizations of scalar elliptic problems as well as for hp-FEM discretizations of the Lame equations of linear elasticity. Moreover, we will consider the Stokes problem as an example of a mixed problem. All preconditioners will be investigated theoretically and numerically in several numerical experiments.