08:30
Error Estimation & Analysis (Part 2)
Chair: Iuliu Sorin Pop
08:30
25 mins
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The Newmark Method and a Space--Time FEM for the Second--Order Wave Equation
Marco Zank
Abstract: For the second--order wave equation, we compare the Newmark Galerkin method with a stabilised space--time finite element method for tensor--product space--time discretisations with piecewise linear, continuous ansatz and test functions leading to an unconditionally stable Galerkin--Petrov scheme, which satisfies a space--time error estimate. We show that both methods require to solve a linear system with the same system matrix. In particular, the stabilised space--time finite element method can be solved sequentially in time as the Newmark Galerkin method. However, the treatment of the right--hand side of the wave equation is different, where the Newmark Galerkin method requires more regularity.
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08:55
25 mins
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Stable variational formulations and discretizations for a kinetic Fokker-Planck equation
Julia Brunken, Mario Ohlberger, Kathrin Smetana
Abstract: In this talk we present a well-posed variational formulation and corresponding stable discretizations for a kinetic Fokker-Planck equation.
The equation describes particle densities in phase space consisting of temporal, spatial, and velocity variables, and is used e.g. for the mesoscopic description of glioma tumor cells.
Since the equation contains diffusive velocity variables as well as advective space and time variables, we cannot use standard theory for parabolic or hyperbolic equations to obtain a well-posed variational formulation.
Instead, we generalize the space-time variational formulation for parabolic equations as defined e.g. in [4]: By incorporating ideas developed for stable variational formulations for first order transport equations [3,2], we generalize the contribution of the time derivative to a higher dimensional advective operator. Thereby, we obtain a well-posed variational formulation on the full space-time-velocity domain based on Bochner-Sobolev-type function spaces similar to the spaces introduced in [1].
Based on this setting, we develop a suitable stable discretization scheme. To tackle the high-dimensionality of the equation, we derive problem adapted bases in the velocity domain.
References
[1] S. Armstrong and J.-C. Mourrat, Variational methods for the kinetic Fokker-Planck equation, arXiv preprint, arXiv:1902.04037 (2019).
[2] J. Brunken, K. Smetana, and K. Urban, (Parametrized) first order transport equations: Realization of optimally stable Petrov–Galerkin methods, SIAM J. Sci. Comput., 41, no. 1, A592–A621 (2019).
[3] W. Dahmen, C. Huang, C. Schwab, and G. Welper, Adaptive Petrov-Galerkin methods for firstorder transport equations, SIAM J. Numer. Anal. 50, no. 5, 2420-2445 (2012).
[4] C. Schwab and R. Stevenson. Space-time adaptive wavelet methods for parabolic evolution problems, Math. Comp. 78, no. 267, 1293–1318 (2009).
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09:20
25 mins
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Study on an adaptive finite element solver for the Cahn-Hilliard equation
Fabian Castelli, Willy Dörfler
Abstract: In this work we present an adaptive matrix-free finite element solver for the Cahn-Hilliard equation modelling phase separation in electrode particles of lithium ion batteries during lithium insertion. We employ an error estimated variable step-variable order time integrator and a regularity estimator for the adaptive mesh refinement. In particular, we propose a matrix-free applicable preconditioner. Numerical experiments demonstrate the importance of adaptive methods and show for our preconditioner practically no dependence of the number of GMRES iterations on the mesh size, even for locally refined meshes.
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09:45
25 mins
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RELIABLE A POSTERIORI ERROR CONTROL WITH MESH ADAPTATIONS FOR COSSERAT ELASTICITY THEORY
Maksim Frolov
Abstract: In this report, we discuss the latest results of implementation of the functional approach [1, 2] to a posteriori error estimation and mesh adaptations for plane problems in Cosserat elasticity. We follow the standard procedure and implement SOLVE-ESTIMATE-MARK-REFINE algorithm in MATLAB with several classical marking strategies (see, for instance, [3, 4]).
Provided by the functional approach, error majorants are always reliable. Results of two different implementation algorithms from [5] and [6] are considered.
REFERENCES
1. S. Repin. A posteriori estimates for partial differential equations. Berlin: de Gruyter, 2008. 316 p.
2. O. Mali, P. Neittaanmaki, S. Repin. Accuracy Verification Methods. Theory and algorithms. Springer, 2014, Vol. 32. 355 p.
3. W. Dorfler. A Convergent Adaptive Algorithm for Poisson’s Equation. SIAM Journal on Numerical Analysis. 1996. No 33(3). P. 1106–1124.
4. R. Verfurth. A review of a posteriori error estimation and adaptive mesh-refinement techniques. Chichester: John Wiley & Sons, Stuttgart: B.G. Teubner, 1996. 127 p.
5. M. Churilova, M. Frolov. Comparison of adaptive algorithms for solving plane problems of classical and Cosserat elasticity. Materials Physics and Mechanics. 2017. No 32(3). P. 370-382.
6. M. Churilova, M. Frolov. A posteriori error estimates for linear problems in Cosserat elasticity. IOP Publishing. Journal of Physics: Conference Series (MMBVPA), Vol. 1158, Issue 2 (2019) 022032, doi 10.1088/1742-6596/1158/2/022032, 8 pages
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