Andreas Schafelner’s research while affiliated with Johannes Kepler University of Linz and other places

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Publications (15)


Goal-oriented adaptive space-time finite element methods for regularized parabolic p-Laplace problems
  • Article

August 2024

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11 Reads

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4 Citations

Computers & Mathematics with Applications

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U. Langer

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A. Schafelner


Figure 5: Example 5.2 (d = 2): Initial space-time mesh (upper left); (x 2 , t)-plane at x 1 = 0.5 of the initial space-time mesh (upper right); space-time mesh after 45 adaptive refinements (lower left);
Figure 6: Example 5.3 (d = 2): Convergence history of the error in the functional as well as efficiency plots, where we additionally included the efficiency of the primal and adjoint parts, respectively.
Figure 7: Example 5.3 (d = 2): From left to right: full space-time mesh, surface mesh of Q I , and the (x 2 , t)-plane at x 1 = 0.5; in its initial configuration (upper row), and after 20 adaptive refinements (lower row); using linear finite elements.
Goal-Oriented Adaptive Space-Time Finite Element Methods for Regularized Parabolic p-Laplace Problems
  • Preprint
  • File available

June 2023

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69 Reads

We consider goal-oriented adaptive space-time finite-element discretizations of the regularized parabolic p-Laplace problem on completely unstructured simplicial space-time meshes. The adaptivity is driven by the dual-weighted residual (DWR) method since we are interested in an accurate computation of some possibly nonlinear functionals at the solution. Such functionals represent goals in which engineers are often more interested than the solution itself. The DWR method requires the numerical solution of a linear adjoint problem that provides the sensitivities for the mesh refinement. This can be done by means of the same full space-time finite element discretization as used for the primal non-linear problems. The numerical experiments presented demonstrate that this goal-oriented, full space-time finite element solver efficiently provides accurate numerical results for different functionals.

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Space-Time Hexahedral Finite Element Methods for Parabolic Evolution Problems

March 2023

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6 Reads

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3 Citations

Lecture Notes in Computational Science and Engineering

Time-stepping methods in combination with some spatial discretization method like the finite element method (FEM) are still the standard approach to the numerical solution of IBVPs like (1); see, e.g., [16]. This time-stepping approach as well as the more recent discontinuous Galerkin, or discontinuous Petrov–Galerkin methods based on time slices or slabs are in principle sequential.


Convergence rates in the mesh-dependent norm ‖(⋅, ⋅)‖h (left); Efficiency indices of the residual indicator and the functional estimator wrt. to the error ϱ‖ ∇x(y−yh) ‖Q2+‖ ∇x(p−ph) ‖Q2 $\sqrt{\varrho \left\| {{\nabla }_{x}}\left( y-{{y}_{h}} \right) \right\|_{Q}^{2}+\left\| {{\nabla }_{x}}\left( p-{{p}_{h}} \right) \right\|_{Q}^{2}}$(right); for different polynomial degrees k. Both plots share the same line styles.
Strong scaling results of the block-AMG preconditioned FGMRES for fixed problem sizes; using linear elements (left), and quadratic elements (right).
Convergence plot of the sum of local error indicators η2(yh,ph)=∑K∈ThηK2(yh,ph) ${{\eta }^{2}}\left( {{y}_{h}},{{p}_{h}} \right)=\sum\nolimits_{K\in {{\mathcal{T}}_{h}}}{{}}\eta _{K}^{2}\left( {{y}_{h}},{{p}_{h}} \right)$for different polynomial degrees k.
Plots of the finite element functions yh, ph, and uh over the mesh, cut at t = 0.3 (upper row); at t = 0.5 (middle row); and at t = 0.7 (bottom row).
Adaptive space-time finite element methods for parabolic optimal control problems

December 2022

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67 Reads

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18 Citations

Journal of Numerical Mathematics

We present, analyze, and test locally stabilized space-time finite element methods on fully unstructured simplicial space-time meshes for the numerical solution of space-time tracking parabolic optimal control problems with the standard L 2 -regularization. We derive a priori discretization error estimates in terms of the local mesh-sizes for shape-regular meshes. The adaptive version is driven by local residual error indicators, or, alternatively, by local error indicators derived from a new functional a posteriori error estimator. The latter provides a guaranteed upper bound of the error, but is more costly than the residual error indicators. We perform numerical tests for benchmark examples having different features. In particular, we consider a discontinuous target in form of a first expanding and then contracting ball in 3d that is fixed in the 4d space-time cylinder.



Space-time hexahedral finite element methods for parabolic evolution problems

March 2021

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12 Reads

We present locally stabilized, conforming space-time finite element methods for parabolic evolution equations on hexahedral decompositions of the space-time cylinder. Tensor-product decompositions allow for anisotropic a priori error estimates, that are explicit in spatial and temporal meshsizes. Moreover, tensor-product finite elements are suitable for anisotropic adaptive mesh refinement strategies provided that an appropriate a posteriori discretization error estimator is available. We present such anisotropic adaptive strategies together with numerical experiments.



Adaptive Space-Time Finite Element Methods for Non-autonomous Parabolic Problems with Distributional Sources

October 2020

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30 Reads

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20 Citations

Computational Methods in Applied Mathematics

We consider locally stabilized, conforming finite element schemes on completely unstructured simplicial space-time meshes for the numerical solution of parabolic initial-boundary value problems with variable coefficients that are possibly discontinuous in space and time. Distributional sources are also admitted. Discontinuous coefficients, non-smooth boundaries, changing boundary conditions, non-smooth or incompatible initial conditions, and non-smooth right-hand sides can lead to non-smooth solutions. We present new a priori and a posteriori error estimates for low-regularity solutions. In order to avoid reduced rates of convergence that appear when performing uniform mesh refinement, we also consider adaptive refinement procedures based on residual a posteriori error indicators and functional a posteriori error estimators. The huge system of space-time finite element equations is then solved by means of GMRES preconditioned by space-time algebraic multigrid. In particular, in the 4d space-time case, simultaneous space-time parallelization can considerably reduce the computational time. We present and discuss numerical results for several examples possessing different regularity features.


Numerical results for adaptive (negative norm) constrained first order system least squares formulations

September 2020

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49 Reads

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6 Citations

Computers & Mathematics with Applications

We perform a followup computational study of the recently proposed space–time first order system least squares ( FOSLS ) method subject to constraints referred to as CFOSLS where we now combine it with the new capability we have developed, namely, parallel adaptive mesh refinement (AMR) in 4D. The AMR is needed to alleviate the high memory demand in the combined space time domain and also allows general (4D) meshes that better follow the physics in space–time. With an extensive set of computational experiments, performed in parallel, we demonstrate the feasibility of the combined space–time AMR approach in both two space plus time and three space plus time dimensions.


Citations (8)


... Finally, we want to show efficiency and reliability for ℎ using an additional assumption. Corollary 3.8 (see [23]). Let us assume that all assumptions of Corollary 3.7 are fulfilled and that there exists a constant ∈ (0, 1) such that ...

Reference:

Mathematical modeling and numerical multigoal-oriented a posteriori error control and adaptivity for a stationary, nonlinear, coupled flow temperature model with temperature dependent density
A posteriori single- and multi-goal error control and adaptivity for partial differential equations
  • Citing Chapter
  • January 2024

Goal-oriented adaptive space-time finite element methods for regularized parabolic p-Laplace problems
  • Citing Article
  • August 2024

Computers & Mathematics with Applications

... Classical norm-based a posteriori error estimation was done for parabolic problems in [21,22,37,39,59,68]. Goal-oriented error estimation of space-time problems was performed B T. Wick thomas.wick@ifam.uni-hannover.de 1 in [10,43,56,57]. ...

Space-Time Hexahedral Finite Element Methods for Parabolic Evolution Problems
  • Citing Chapter
  • March 2023

Lecture Notes in Computational Science and Engineering

... Further investigations on space-time finite element methods have been done in [18,19,42], where numerical results with an adaptive algorithm are presented. The advantages of the variational time discretization are the natural integration with the variational space discretization and the natural capture of coupled problems and nonlinearities. ...

Adaptive space-time finite element methods for parabolic optimal control problems

Journal of Numerical Mathematics

... In this setting, a moving domain can conveniently be captured by the space-time mesh. While, at the first glance, the method comes with the challenge of higher-dimensional linear systems to be solved, it allows for both parallelization [18] and adaptivity [26,40] not only in space or time, but also in space-time. Moreover, in the context of optimization problems with partial differential equations as constraint, and involving an adjoint state which is directed backward in time, space-time methods allow for an additional level of parallelism by solving the coupled system for the state and the adjoint in parallel [27]. ...

Adaptive Space-Time Finite Element Methods for Non-autonomous Parabolic Problems with Distributional Sources
  • Citing Article
  • October 2020

Computational Methods in Applied Mathematics

... The problem of developing an efficient solver for space-time formulations is important for many applications. The number of papers and citations in the field of space-time formulations is growing exponentially (Web of Science search, space-time formulation), with some novel examples [5][6][7][8][9][10][11][12][13]. The direct solvers for space-time formulations are costly [14]. ...

Numerical results for adaptive (negative norm) constrained first order system least squares formulations

Computers & Mathematics with Applications

... The systems arising from space-time finite element discretizations are well-suited for solution by STMG methods. However, most work is based on algebraic multigrid methods [62,43,44]. Geometric STMG methods have first been addressed in [34] and later in [30,39] as well as in the references therein. ...

Space-Time Finite Element Methods for Parabolic Initial-Boundary Value Problems with Non-smooth Solutions
  • Citing Chapter
  • February 2020

Lecture Notes in Computer Science

... This conclusion is further supported by the error computed from (33) using δ = 0.45 and k = 2, yielding a value of 0.149, which represents a 14.77% improvement compared to the corresponding error reported in Table 2 for the analytical solution with constant boundary conditions. Building on these results, a more comprehensive approach to designing drug delivery systems that undergo significant geometric changes and time-dependent boundary conditions should incorporate space-time FE methods (see, e.g., [43,44]). Future work will explore this approach by incorporating additional parameters into the general form of diffusion coefficient presented in Remark 1, with the aim of capturing effects such as evaporation within the composite. ...

Space-Time Finite Element Methods for Parabolic Evolution Problems with Variable Coefficients
  • Citing Chapter
  • June 2019

Lecture Notes in Computational Science and Engineering