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Author: Eigel, Martin × Subject: adaptivity ×

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Now showing 1 - 5 of 5
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    Functional a posteriori error estimation for stationary reaction-convection-diffusion problems
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Eigel, Martin; Samrowski, Tatiana
    A functional type a posteriori error estimator for the finite element discretisation of the stationary reaction-convection-diffusion equation is derived. In case of dominant convection, the solution for this class of problems typically exhibits boundary layers and shock-front like areas with steep gradients. This renders the accurate numerical solution very demanding and appropriate techniques for the adaptive resolution of regions with large approximation errors are crucial. Functional error estimators as derived here contain no mesh-dependent constants and provide guaranteed error bounds for any conforming approximation. To evaluate the error estimator, a minimisation problem is solved which does not require any Galerkin orthogonality or any specific properties of the employed approximation space. Based on a set of numerical examples, we assess the performance of the new estimator. It is observed that it exhibits a good efficiency also with convection-dominated problem settings.
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    An adaptive multi level Monte-Carlo method with stochastic bounds for quantities of interest in groundwater flow with uncertain data
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Eigel, Martin; Merdon, Christian; Neumann, Johannes
    The focus of this work is the introduction of some computable a posteriori error control to the popular multilevel Monte Carlo sampling for PDE with stochastic data. We are especially interested in applications in the geosciences such as groundwater flow with rather rough stochastic fields for the conductive permeability. With a spatial discretisation based on finite elements, a goal functional is defined which encodes the quantity of interest. The devised goal-oriented error estimator enables to determine guaranteed a posteriori error bounds for this quantity. In particular, it allows for the adaptive refinement of the mesh hierarchy used in the multilevel Monte Carlo simulation. In addition to controlling the deterministic error, we also suggest how to treat the stochastic error in probability. Numerical experiments illustrate the performance of the presented adaptive algorithm for a posteriori error control in multilevel Monte Carlo methods. These include a localised goal with problem-adapted meshes and a slit domain example. The latter demonstrates the refinement of regions with low solution regularity based on an inexpensive explicit error estimator in the multilevel algorithm.
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    Adaptive non-intrusive reconstruction of solutions to high-dimensional parametric PDEs
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Eigel, Martin; Farchmin, Nando; Heidenreich, Sebastian; Trunschke, Philipp
    Numerical methods for random parametric PDEs can greatly benefit from adaptive refinement schemes, in particular when functional approximations are computed as in stochastic Galerkin and stochastic collocations methods. This work is concerned with a non-intrusive generalization of the adaptive Galerkin FEM with residual based error estimation. It combines the non-intrusive character of a randomized least-squares method with the a posteriori error analysis of stochastic Galerkin methods. The proposed approach uses the Variational Monte Carlo method to obtain a quasi-optimal low-rank approximation of the Galerkin projection in a highly efficient hierarchical tensor format. We derive an adaptive refinement algorithm which is steered by a reliable error estimator. Opposite to stochastic Galerkin methods, the approach is easily applicable to a wide range of problems, enabling a fully automated adjustment of all discretization parameters. Benchmark examples with affine and (unbounded) lognormal coefficient fields illustrate the performance of the non-intrusive adaptive algorithm, showing best-in-class performance
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    Robust equilibration a posteriori error estimation for convection-diffusion-reaction problems
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Eigel, Martin; Merdon, Christian
    We study a posteriori error estimates for convection-diffusion-reaction problems with possibly dominating convection or reaction and inhomogeneous boundary conditions. For the conforming FEM discretisation with streamline diffusion stabilisation (SDM), we derive robust and efficient error estimators based on the reconstruction of equilibrated fluxes in an admissible discrete subspace of H(div, Omega). Error estimators of this type have become popular recently since they provide guaranteed error bounds without further unknown constants. The estimators can be improved significantly by some postprocessing and divergence correction technique. For an extension of the energy norm by a dual norm of some part of the differential operator, complete independence from the coefficients of the problem is achieved. Numerical benchmarks illustrate the very good performance of the error estimators in the convection dominated and the singularly perturbed cases.
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    Reliable averaging for the primal variable in the Courant FEM and hierarchical error estimators on red-refined meshes
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Carstensen, Carsten; Eigel, Martin
    A hierarchical a posteriori error estimator for the first-order finite element method (FEM) on a red-refined triangular mesh is presented for the 2D Poisson model problem. Reliability and efficiency with some explicit constant is proved for triangulations with inner angles smaller than or equal to π/2 . The error estimator does not rely on any saturation assumption and is valid even in the pre-asymptotic regime on arbitrarily coarse meshes. The evaluation of the estimator is a simple post-processing of the piecewise linear FEM without any extra solve plus a higher-order approximation term. The results also allows the striking observation that arbitrary local averaging of the primal variable leads to a reliable and efficient error estimation. Several numerical experiments illustrate the performance of the proposed a posteriori error estimator for computational benchmarks.