In the area of numerical analysis, research is conducted on anisotropic finite elements, on the discretization of optimal control problems, and on eigenvalues of differential operators.

 

Anisotropic Finite Elements

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The use of anisotropic finite elements is particularly efficient when the sought solution has anisotropic features such as edge singularities or boundary layers. In this research field, IMCS mainly investigates discretization errors and derives both a priori and a posteriori error estimates.

Contacts at IMCS
Key Publications
  • Apel, T. (1999): Anisotropic finite elements: Local estimates and applications. Teubner, Stuttgart.
  • Apel, T., Lombardi, A. L., Winkler, M. (2014): Anisotropic mesh refinement in polyhedral domains: error estimates with data in L2(Omega). ESAIM: Math. Model. Numer. Anal. 48(4): 1117-1145.
  • Apel, T.; Kempf, V. (2020): Brezzi--Douglas--Marini interpolation of any order on anisotropic triangles and tetrahedra, SIAM Journal on Numerical Analysis, 58(3): 1696-1718.
Current Projects
Completed Projects

 

Discretization of Optimal Control Problems

solution_l2.pngIn optimal control with partial differential equations, the data are determined such that the solution has desired properties. At IMCS, we deal with the discretization of such tasks, in particular, we adapt the discretization to the problem and estimate the discretization error.

Contacts at IMCS
Key Publications
  • Apel, T., Pfefferer, J., Rösch, A. (2014): Locally refined meshes in optimal control for elliptic partial differential equations - an overview. In: G. Leugering, P. Benner, S. Engell, A. Griewank, H. Harbrecht, M. Hinze, R. Rannacher, and S. Ulbrich (eds.): Trends in PDE Constrained Optimization. International Series of Numerical Mathematics 165. Birkhäuser, Basel, pp. 285-302.
  • Apel, T., Mateos, M., Pfefferer, J., Rösch, A. (2015): On the regularity of the solutions of Dirichlet optimal control problems in polygonal domains, SIAM J. Control Optim. 53, 3620–3641.
  • Apel, T., Pfefferer, J., Rogovs, S., Winkler, M. (2018): L∞-error estimates for Neumann boundary value problems on graded meshes. IMA J. Numer. Anal. 40, 474–497.
Current Projects
Completed Projects

 

Eigenvalues of Differential Operators

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The solution of differential equations can often be represented by a series, where one factor of the series elements is represented by eigenfunctions of differential operators. Consider, for example, natural modes of oscillation in vibration problems. At IMCS we develop adapted discretizations for such problems and prove their efficiency.

Contacts am IMCS
Key publications
  • Apel, T., Sändig, A.-M., Solov'ev, S.I. (2002): Computation of 3D vertex singularities for linear elasticity: Error estimates for a finite element method on graded meshes. Math. Model. Numer. Anal., 36:1043--1070.
  • Apel, T., Mehrmann, V., Watkins, D. (2002): Numerical solution of large scale structured polynomial or rational eigenvalue problems. In F. Cucker, R. DeVore, P. Olver, and E. Süli, editors, Foundations of Computational Mathematics, Minneapolis 2002, volume 312 of Lecture Note Series, Cambridge, 2004. London Mathematical Society, Cambridge University Press.
  • Apel,T., Pester, C. (2005): Clement-type interpolation on spherical domains - interpolation error estimates and application to a posteriori error estimation. IMA Journal of Numerical Analysis, 25, No.2, 310-336.
Current Projects
Completed Projects