Isogeometric Method for Partial Differential Equations.

The research team for this project, which was funded by Austrian Science Fund during July 2009 to November 2013, consisted the PI (S.K. Tomar) and two PhD students K.P.S. Gahalaut and S.K. Kleiss. The main developments in this project, according to the stated objectives, are listed below.

  1. Fast solvers for linear system of equations: This topic was a major focus during this project.
    1. We began with graph theory based preconditioners, a relatively not-so-developed topic but which has been shown to be very effective in certain cases. Unfortunately, it was found, see the preliminary study in [1], that this approach is not suitable for IGA, particularly, due to large overlaps of basis functions (and thus resulting more connections in the matrices), and was not pursued further.
    2. An interesting approach, particularly, well suited when the problem domain can be decomposed in to several sub-domains, is the tearing and interconnecting approach, well studied in finite element methods and boundary element methods. The resulting isogeometric tearing and interconnecting (IETI) method, and associated local refinement strategies were presented in [2].
    3. Since the geometry is exactly represented in IGA, multigrid methods are a natural choice for the iterative solution of the linear system of equations. In particular, as long as the numerical solution has full regularity (without any corner singularities or the jumps in the PDE operator), these methods offer the linear order of convergence. As a first study on the topic, we proved necessary theoretically results and provided exhaustive numerical results in [3].
    4. The results of multigrid methods in [3] were excellent for low polynomial degree p. However, particularly for three-dimensional problems, larger p clearly reflected the limitation of this methodology. We then constructed algebraic multilevel preconditioners. The proposed method delivered, apart from h-independence, almost p-independence [4]. Apart from the construction of the required coarse grid, hierarchical complementary operators, and supporting numerical results, we also provided explicit representation formulas for B-splines.
      Since the overlapping support of basis functions increases with p, this posed serious challenges in theoretically proving the robustness of the method, particularly for high p. This is still an open problem and it is intended to be addressed in near future.
    5. When dealing with iterative solvers, the condition number of the resulting matrices (from the discretization) is an important property. Therefore, in [6], for the mass and stiffness matrices we studied the condition number estimates computationally as well as with rigorous theoretical estimates. Apart from h– refinement, we have also focused on p– and r– refinements.
  2. Guaranteed and sharp a posteriori error estimates: In general situations, the reliability and efficiency of other a posteriori error estimation methods found in the literature mostly depend on undetermined constants, which is not suitable for quality assurance purposes. In [5], we proposed functional-type a posteriori error estimates for the isogeometric discretization of elliptic problems. These estimates, derived on functional grounds, provide guaranteed and sharp upper bounds of the exact error in the energy norm. Moreover, since these estimates do not contain any unknown / generic constants, they are fully computable, and thus provide quantitative information on the error. By exploiting the properties of non-uniform rational B-splines, we presented efficient computation of these error estimates. In the PhD thesis of S.K. Kleiss, lower bounds of the exact error is also discussed.

 

Bibliography

  1. Condition number study of graph theory based preconditioners for isogeometric discretization of Poisson equation (with K.P.S. Gahalaut). RICAM Report, 2012-14.
  2. IETI – Isogeometric Tearing and Interconnecting (with B.J. Juettler, S.K. Kleiss and C. Pechstein). Comput. Methods Appl. Mech. Engrg.247-248, 201-215, 2012.
  3. Multigrid Methods for Isogeometric Discretization (with K.P.S. Gahalaut and J. Kraus). Comput. Methods Appl. Mech. Engrg., 253, 413-425, 2013.
  4. Algebraic multilevel preconditioning in isogeometric analysis: Construction and numerical studies (with K.P.S. Gahalaut and J. Kraus). Comput. Methods Appl. Mech. Engrg.266, 40-56, 2013.
  5. Guaranteed and sharp a posteriori error estimates in isogeometric analysis (with S.K. Kleiss). Published onlineComput. Math. Appl.. Also available at arXiv:1304.7712.
  6. Condition number estimates for matrices arising in the isogeometric discretizations (with K.P.S. Gahalaut). In revision (Submitted for publication in October 2012). Available as RICAM Report, 2012-23.