MS51: Space and time adaptation for PDEs: From theory to practice (Part 2)
Chair: Maurizio Falcone
Aposteriori analysis of hp-discontinuous Galerkin timestepping methods for fully discretized parabolic problems
Omar Lakkis, Emmanuil H. Georgoulis, Thomas P Wihler
Abstract: We derive aposteriori error bounds in time-maximum-space-squared-sums
and time-mean-squares-of-spatial-energy norms for a class of
fully-discrete methods for linear parabolic partial differential
equations (PDEs) on the space-time domain based on hp-version
discontinuous Galerkin time-stepping scheme combined with conforming
spatial Galerkin finite element method. The proofs are based on a
novel space-time reconstruction, which combines the elliptic
reconstruction of Georgoulis, Lakkis & Virtanen (2011), Lakkis &
Makridakis (2006), and Makridakis & Nochetto (2003) and the time
reconstruction for discontinuous time-Galerkin schemes of Makridakis &
Nochetto (2006) and Schötzau & Wihler (2010) into a novel tool which
allows for the user's preferred choice of aposteriori error estimates
in space and a fine quantification of mesh-change effects.
Adaptive finite element approaches for simulations of Newman-type Lithium ion battery models
Thomas Carraro, Sven E. Wetterauer
Abstract: Diffusion and heterogeneous reaction processes characterize electrochemical systems like fuel cells, oxygen capturing membranes and Lithium ion batteries (LIB).
When these processes occur in complex microstructures the physical phenomena have a multiscale character. From the practical point of view usually a macroscopic rather than microscopic quantity is of interest, e.g., the flux over a surface, i.e. the current in case of electrochemical systems.
The direct computation of this quantity is typically beyond the computational capacities even of high performace computer systems. Therefore, a reduction of the problem is necessary.
A well established homogenized LIB model is the so called Newmann-type model also denoted P2D model, meaning that it is pseudo two-dimensional. It is a multiscale model defined in different material phases with a formulation that contains microscopic and macroscopic components.
We present a dual based adaptive finite element method for its numerical solution.
For the computation of the effective diffusion coefficients on the complex microstructure of the electrode we present a 3D adaptive cut-cell implementation based on a level set method and a goal oriented error estimator.
Anisotropic mesh adaptation for image segmentation via a variational approach
Alberto Silvio Chiappa, Stefano Micheletti, Riccardo Peli, Simona Perotto
Abstract: Many applications in image processing require the identification of objects and boundaries, such as lines and curves, collectively referred to as segments, e.g., X-ray based inspection techniques, satellite imagery, medical diagnostics, texture and facial recognition, trajectory planning.
With the aim of devising fast and reliable algorithms to accomplish this task, we focus on a variational approach to image segmentation based on the Ambrosio-Tortorelli functional . To make the procedure more effective with respect to standard algorithms, we perform a finite element approximation of the Ambrosio-Tortorelli functional on a triangular adapted mesh able to follow exactly the contours present in the images.
This goal is reached via a rigorous a posteriori error analysis based on anisotropic information. The extensive numerical investigation shows that the proposed adaptive method is reliable on both synthetic and realimages. We succeed in effectively segmenting even complex images as those characterizing medical applications. This is
obtained on quite coarse meshes in contrast to the original pixel representation, or to isotropically adapted grids.
 A.S. Chiappa, S. Micheletti, R. Peli, S. Perotto, Mesh adaptation-aided image segmentation, Commun Nonlinear Sci Numer Simulat 74 (2019) 147-166
Adaptive Filtered Schemes for First Order Evolutive Hamilton-Jacobi Equations
Maurizio Falcone, Giulio Paolucci, Silvia Tozza
Abstract: The accurate numerical solution of Hamilton-Jacobi (HJ) equations is a challenging topic of growing importance in many fields of application, e.g. control theory, KAM theory, image processing and material science.
This is a delicate issue due to the lack of regularity of viscosity solutions , so the construction of high-order methods can be rather difficult. In fact simple monotone schemes are at most first order accurate so monotonicity should be abandoned and the proof of high-order convergence becomes very challenging. Several methods have been proposed (e.g. ENO and WENO schemes ) but a precise convergence result is still missing and their implementation is often tricky .
More recently, the simple idea of filtered schemes for HJ equations has been proposed. They are obtained coupling two different schemes: one is monotone (and convergent) and the other is high-order accurate. The filtered scheme switches from one scheme to the other according to a filter function and a switching parameter, this feature is crucial to prove high-order convergence where the solution is regular. Then it seems natural to adapt the choice of this parameter to the regularity of the solution in the cell via a smoothness indicator improving the standard filtered scheme (where this parameter is constant ) by an adaptive and automatic choice at every iteration.
Here we introduce a smoothness indicator to select the regions where we have to update the regularity threshold and we adapt the parameter in time and space.
We present a general convergence result and some error estimates for the new adaptive filtered scheme in 1D . Finally, we show a 2D application to the segmentation problem in image processing .
 G. Barles, Solutions de viscositè des equations de Hamilton-Jacobi, Springer Verlag, 1994.
 O. Bokanowski, M. Falcone, S. Sahu, An efficient filtered scheme for some first order Hamilton-Jacobi-Bellman equations, SIAM Journal on Scientific Computing 38:1 (2016), A171–A195.
 M. Falcone and R. Ferretti, Semi-Lagrangian Approximation Schemes for Linear and Hamilton-Jacobi Equations, SIAM, 2013.
 M. Falcone, G. Paolucci, and S. Tozza, A High-Order Scheme for Image Segmentation via a modified Level-Set method. 2018, submitted.
 M. Falcone, G. Paolucci, and S. Tozza, Convergence of Adaptive Filtered schemes for first order evolutive Hamilton-Jacobi equations. 2018, submitted, arXiv: 1812.02140
 G. Jiang, D.-P. Peng, Weighted ENO schemes for Hamilton-Jacobi equations, SIAM Journal on Scientific Computing 21 (2000), 2126–2143.