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Coquel, M. Gutnic, P. Helluy, F.

37. Runge-Kutta Method - Problem#2 - Complete Concept

Rohde, and N. Seguin, Eds. Corli and C. Nolte, and M. Dedner, B. Flemisch, and R. Springer, Dihlmann, S. Kaulmann, and B. Kraus, and C. Drohmann, B. Haasdonk, and M.

Professor L N Trefethen FRS: Publications

For both settings, model order reduction by the reduced basis approach is a suitable means to reduce computational time. The method is based on a projection of an underlying high dimensional nu- merical scheme onto a low dimensional function space. In this contribution, a new software framework is introduced that allows fast development of reduced schemes for a large class of discretizations of evolution equations implemented in DUNE.

The functionality of the framework is exemplified for a finite volume approximation of an instationary linear convection-diffusion problem. An a posteriori error estimate for the resulting reduced basis method is derived and analyzed numerically. We introduce a new algorithm, the PODEI-greedy algorithm, which constructs the reduced basis spaces for the empirical interpolation and for the numerical scheme in a synchronized way. The approach is applied to nonlinear parabolic and hyperbolic equations based on explicit or implicit finite volume discretizations.

We show that the resulting reduced scheme is able to capture the evolution of both smooth and discontinuous solutions.


In case of symmetries of the problem, the approach realizes an automatic and intuitive space-compression or even space-dimensionality reduction. We perform empirical investigations of the error convergence and run-times.

Numerical methods for fluids

In all cases we obtain a good run-time acceleration. Engel and C. Feistauer and A. In this article, we consider a Dirichlet and a mixed boundary value problem for a linear elliptic equation in a polygon. It is well known that the first derivatives of the solutions develop singularities near reentrant corner points or points where the boundary conditions change. On the basis of the regularity results formulated in Sobolev--Slobodetskii spaces and weighted spaces of Kondratiev type, we prove error estimates of higher order for DGFE solutions using a suitable graded mesh refinement near boundary singular points.

The main tools are as follows: regularity investigation for the exact solution relying on general results for elliptic boundary value problems, error analysis for the interpolation in Sobolev--Slobodetskii spaces, and error estimates for DGFE solutions on special graded refined meshes combined with estimates in weighted Sobolev spaces. Fornasier, Y. Kim, A.

Convergence of a Mimetic Finite Difference Method for Static Diffusion Equation

Langer, and C. Giesselmann and M. Klar, G. Leugering, and V. Hoher, P. Schindler, S. Schleper, and S.

Jackson, I. Rybak, R. Helmig, W. Proceedings and Surveys , Marseille, France, , no. We consider the fronts in this contribution as sharp interfaces and propose a novel multiscale approach. First numerical experiments in three space dimensions underline the overall performance of the method. Kelkel and C. Kissling, R. It is solely based on reliable and efficient error estimators for the corresponding linear state and adjoint equations. We show that the sum of these estimators gives a reliable and efficient estimator for the optimal control problem.

Kohr, G. Raja Sekhar, E. Ului, and W. Kreuzer, C. Schmidt, and K. It is based on adaptive finite elements in space and the implicit Euler discretization in time with adaptive time-step sizes. We prove that, given a positive tolerance for the error, the adaptive algorithm reaches the final time with a space—time error between continuous and discrete solution that is below the given tolerance. Numerical experiments reveal a more than competitive performance of our algorithm ASTFEM adaptive space—time finite element method. Langer, M. Schanz, O. Steinbach, and W. Berlin: Springer, , p. Richter et al. Rohde and F.

Ruiner, J. Fehr, B. We introduce finite element spaces, discuss local refinement of simplical grids, the assemblage and structure of the discrete linear system, the computation of the error estimator, and common adaptive strategies. The mathematical discussion naturally leads to appropriate data structures and efficient algorithms for the implementation. The theoretical part is complemented by exercises giving an introduction to the implementation of solvers for linear and nonlinear problems in the adaptive finite element toolbox ALBERTA.

Steinhorst and A. Therefore, it is important to develop nondestructive techniques for testing whether cracks exist inside the device, and if they exist, what the position and size of these cracks are. In this paper, we focus on the inverse method of measurements of boundary data for anisotropic elastic fields under different loads in order to detect cracks. We draw on the paper of Andrieux et al Inverse Problems 15 , where a reciprocity principle is used for the detection of plane cracks in isotropic linear elastic materials.

Utilizing that the setup and significant parts of the analysis by Andrieux et al are formulated generally and remain valid in anisotropic materials, we generalize this method for two- and three-dimensional transversely anisotropic elastic devices and demonstrate it by numerical experiments. We start from a cracked domain and generate boundary data as artificial measurements by solving numerically a forward problem.

The inverse computations show good agreement between the simulated crack and the original one. Waldherr and B.