scholarly journals A posteriori error control for the finite cell method

PAMM ◽  
2019 ◽  
Vol 19 (1) ◽  
Author(s):  
Paolo Di Stolfo ◽  
Alexander Düster ◽  
Stefan Kollmannsberger ◽  
Ernst Rank ◽  
Andreas Schröder
Keyword(s):  
Error Control ◽  
Posteriori Error ◽  
A Posteriori ◽  
Finite Cell Method ◽  
A Posteriori Error ◽  
Cell Method ◽  
Finite Cell

scholarly journals Reliable Residual-Based Error Estimation for the Finite Cell Method

2021 ◽  
Vol 87 (1) ◽  
Author(s):  
Paolo Di Stolfo ◽  
Andreas Schröder
Keyword(s):  
Error Control ◽  
Extension Theorem ◽  
Mesh Size ◽  
Neumann Boundary ◽  
Posteriori Error ◽  
Error Estimator ◽  
Finite Cell Method ◽  
Polynomial Degree ◽  
Cell Method ◽  
Finite Cell

AbstractIn this work, the reliability of a residual-based error estimator for the Finite Cell method is established. The error estimator is suitable for the application of hp-adaptive finite elements and allows for Neumann boundary conditions on curved boundaries. The reliability proof of the error estimator relies on standard arguments of residual-based a posteriori error control, but includes several modifications with respect to the error contributions associated with the volume residuals as well as the jumps across inner edges and Neumann boundary parts. Important ingredients of the proof are Stein’s extension theorem and a modified trace theorem which estimates the norm of the trace on (curved) boundary parts in terms of the local mesh size and polynomial degree. The efficiency of the error estimator is also considered by discussing an artificial example which yields an efficiency index depending on the mesh-family parameter h. Numerical experiments on more realistic domains, however, suggest global efficiency with the occurrence of a large overestimation on only few cut elements. In the experiments the reliability of the error estimator is demonstrated for h- and p-uniform as well as for hp-geometric and h-adaptive refinements driven by the error estimator. The practical applicability of the error estimator is also studied for a 3D problem with a non-smooth solution.

scholarly journals Adaptive regularization, linearization, and discretization and a posteriori error control for the two-phase Stefan problem

2014 ◽  
Vol 84 (291) ◽  
pp. 153-186 ◽  
Author(s):  
Daniele A. Di Pietro ◽  
Martin Vohralík ◽  
Soleiman Yousef
Keyword(s):  
Stefan Problem ◽  
Error Control ◽  
Posteriori Error ◽  
A Posteriori ◽  
Two Phase ◽  
Adaptive Regularization ◽  
A Posteriori Error

scholarly journals Functional A Posteriori Error Control for Conforming Mixed Approximations of Coercive Problems with Lower Order Terms

2016 ◽  
Vol 16 (4) ◽  
pp. 609-631 ◽  
Author(s):  
Immanuel Anjam ◽  
Dirk Pauly
Keyword(s):  
Error Control ◽  
A Posteriori Error Estimates ◽  
Reaction Diffusion ◽  
Posteriori Error ◽  
Diffusion Problem ◽  
Mixed Formulation ◽  
A Posteriori ◽  
A Posteriori Error ◽  
Linear Partial Differential Equations ◽  
Primal And Dual

AbstractThe results of this contribution are derived in the framework of functional type a posteriori error estimates. The error is measured in a combined norm which takes into account both the primal and dual variables denoted by x and y, respectively. Our first main result is an error equality for all equations of the class ${\mathrm{A}^{*}\mathrm{A}x+x=f}$ or in mixed formulation ${\mathrm{A}^{*}y+x=f}$, ${\mathrm{A}x=y}$, where the exact solution $(x,y)$ is in $D(\mathrm{A})\times D(\mathrm{A}^{*})$. Here ${\mathrm{A}}$ is a linear, densely defined and closed (usually a differential) operator and ${\mathrm{A}^{*}}$ its adjoint. In this paper we deal with very conforming mixed approximations, i.e., we assume that the approximation ${(\tilde{x},\tilde{y})}$ belongs to ${D(\mathrm{A})\times D(\mathrm{A}^{*})}$. In order to obtain the exact global error value of this approximation one only needs the problem data and the mixed approximation itself, i.e., we have the equality$\lvert x-\tilde{x}\rvert^{2}+\lvert\mathrm{A}(x-\tilde{x})\rvert^{2}+\lvert y-% \tilde{y}\rvert^{2}+\lvert\mathrm{A}^{*}(y-\tilde{y})\rvert^{2}=\mathcal{M}(% \tilde{x},\tilde{y}),$where ${\mathcal{M}(\tilde{x},\tilde{y}):=\lvert f-\tilde{x}-\mathrm{A}^{*}\tilde{y}% \rvert^{2}+\lvert\tilde{y}-\mathrm{A}\tilde{x}\rvert^{2}}$ contains only known data. Our second main result is an error estimate for all equations of the class ${\mathrm{A}^{*}\mathrm{A}x+ix=f}$ or in mixed formulation ${\mathrm{A}^{*}y+ix=f}$, ${\mathrm{A}x=y}$, where i is the imaginary unit. For this problem we have the two-sided estimate$\frac{\sqrt{2}}{\sqrt{2}+1}\mathcal{M}_{i}(\tilde{x},\tilde{y})\leq\lvert x-% \tilde{x}\rvert^{2}+\lvert\mathrm{A}(x-\tilde{x})\rvert^{2}+\lvert y-\tilde{y}% \rvert^{2}+\lvert\mathrm{A}^{*}(y-\tilde{y})\rvert^{2}\leq\frac{\sqrt{2}}{% \sqrt{2}-1}\mathcal{M}_{i}(\tilde{x},\tilde{y}),$where ${\mathcal{M}_{i}(\tilde{x},\tilde{y}):=\lvert f-i\tilde{x}-\mathrm{A}^{*}% \tilde{y}\rvert^{2}+\lvert\tilde{y}-\mathrm{A}\tilde{x}\rvert^{2}}$ contains only known data. We will point out a motivation for the study of the latter problems by time discretizations or time-harmonic ansatz of linear partial differential equations and we will present an extensive list of applications including the reaction-diffusion problem and the eddy current problem.

scholarly journals Dual weighted residual error estimation for the finite cell method

2019 ◽  
Vol 27 (2) ◽  
pp. 101-122 ◽  
Author(s):  
Paolo Di Stolfo ◽  
Andreas Rademacher ◽  
Andreas Schröder
Keyword(s):  
Error Estimation ◽  
Error Control ◽  
Combined Treatment ◽  
Adaptive Strategy ◽  
Two Dimensional ◽  
Weighted Residual Method ◽  
Finite Cell Method ◽  
Cell Method ◽  
Quadrature Error ◽  
Finite Cell

Abstract The paper presents a goal-oriented error control based on the dual weighted residual method (DWR) for the finite cell method (FCM), which is characterized by an enclosing domain covering the domain of the problem. The error identity derived by the DWR method allows for a combined treatment of the discretization and quadrature error introduced by the FCM. We present an adaptive strategy with the aim to balance these two error contributions. Its performance is demonstrated for several two-dimensional examples.

Pointwise a posteriori error control for elliptic obstacle problems

2003 ◽  
Vol 95 (1) ◽  
pp. 163-195 ◽  
Author(s):  
Ricardo H. Nochetto ◽  
Kunibert G. Siebert ◽  
Andreas Veeser
Keyword(s):  
Error Control ◽  
Posteriori Error ◽  
Obstacle Problems ◽  
A Posteriori ◽  
A Posteriori Error

scholarly journals A Posteriori Error Control for Fully Discrete Crank--Nicolson Schemes

2012 ◽  
Vol 50 (6) ◽  
pp. 2845-2872 ◽  
Author(s):  
E. Bänsch ◽  
F. Karakatsani ◽  
Ch. Makridakis
Keyword(s):  
Error Control ◽  
Posteriori Error ◽  
A Posteriori ◽  
A Posteriori Error ◽  
Fully Discrete ◽  
Crank Nicolson

On a posteriori error control for the Allen-Cahn problem

2013 ◽  
Vol 37 (2) ◽  
pp. 173-179 ◽  
Author(s):  
Emmanuil H. Georgoulis ◽  
Charalambos Makridakis
Keyword(s):  
Error Control ◽  
Posteriori Error ◽  
A Posteriori ◽  
A Posteriori Error
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