Details
| Original language | English |
|---|---|
| Article number | 118034 |
| Journal | Computer Methods in Applied Mechanics and Engineering |
| Volume | 443 |
| Early online date | 13 May 2025 |
| Publication status | E-pub ahead of print - 13 May 2025 |
Abstract
Goal-oriented a posteriori error estimation is crucial for solving partial differential equations (PDEs) efficiently and reliably. The Virtual Element Method (VEM) shows promise in this context due to its ability to handle general polygonal elements, eliminating the need for special treatment of hanging nodes. However, a suitable framework for goal-oriented error estimation in VEM has not been developed so far. This work addresses this gap by deriving an appropriate estimator formulation for linear PDEs using VEM. We tackle two key challenges for first-order Virtual Elements: approximating virtual basis functions within elements and efficiently approximating the exact adjoint solution, where standard methods used for finite element approximations are not suitable. To overcome these challenges, we introduce new techniques, including the Gauss-Point Reconstruction Method (GPRM). Our theoretical developments are verified through diverse numerical experiments, demonstrating their correctness and effectiveness. We further showcase the practical utility of our framework through its application to adaptive mesh refinement, which enhances solution accuracy while optimizing computational resources. This work lays the foundation for extending goal-oriented error estimation to more complex problems using VEM.
Keywords
- Adaptivity, Goal-Oriented DWR error estimation, Virtual Elements
ASJC Scopus subject areas
- Engineering(all)
- Computational Mechanics
- Engineering(all)
- Mechanics of Materials
- Engineering(all)
- Mechanical Engineering
- Physics and Astronomy(all)
- General Physics and Astronomy
- Computer Science(all)
- Computer Science Applications
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In: Computer Methods in Applied Mechanics and Engineering, Vol. 443, 118034, 01.08.2025.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Goal-oriented dual-weighted residual error estimation for the Virtual Elements Method
AU - Sellmann, C.
AU - Junker, P.
N1 - Publisher Copyright: © 2025 The Authors
PY - 2025/5/13
Y1 - 2025/5/13
N2 - Goal-oriented a posteriori error estimation is crucial for solving partial differential equations (PDEs) efficiently and reliably. The Virtual Element Method (VEM) shows promise in this context due to its ability to handle general polygonal elements, eliminating the need for special treatment of hanging nodes. However, a suitable framework for goal-oriented error estimation in VEM has not been developed so far. This work addresses this gap by deriving an appropriate estimator formulation for linear PDEs using VEM. We tackle two key challenges for first-order Virtual Elements: approximating virtual basis functions within elements and efficiently approximating the exact adjoint solution, where standard methods used for finite element approximations are not suitable. To overcome these challenges, we introduce new techniques, including the Gauss-Point Reconstruction Method (GPRM). Our theoretical developments are verified through diverse numerical experiments, demonstrating their correctness and effectiveness. We further showcase the practical utility of our framework through its application to adaptive mesh refinement, which enhances solution accuracy while optimizing computational resources. This work lays the foundation for extending goal-oriented error estimation to more complex problems using VEM.
AB - Goal-oriented a posteriori error estimation is crucial for solving partial differential equations (PDEs) efficiently and reliably. The Virtual Element Method (VEM) shows promise in this context due to its ability to handle general polygonal elements, eliminating the need for special treatment of hanging nodes. However, a suitable framework for goal-oriented error estimation in VEM has not been developed so far. This work addresses this gap by deriving an appropriate estimator formulation for linear PDEs using VEM. We tackle two key challenges for first-order Virtual Elements: approximating virtual basis functions within elements and efficiently approximating the exact adjoint solution, where standard methods used for finite element approximations are not suitable. To overcome these challenges, we introduce new techniques, including the Gauss-Point Reconstruction Method (GPRM). Our theoretical developments are verified through diverse numerical experiments, demonstrating their correctness and effectiveness. We further showcase the practical utility of our framework through its application to adaptive mesh refinement, which enhances solution accuracy while optimizing computational resources. This work lays the foundation for extending goal-oriented error estimation to more complex problems using VEM.
KW - Adaptivity
KW - Goal-Oriented DWR error estimation
KW - Virtual Elements
UR - http://www.scopus.com/inward/record.url?scp=105004699465&partnerID=8YFLogxK
U2 - 10.1016/j.cma.2025.118034
DO - 10.1016/j.cma.2025.118034
M3 - Article
AN - SCOPUS:105004699465
VL - 443
JO - Computer Methods in Applied Mechanics and Engineering
JF - Computer Methods in Applied Mechanics and Engineering
SN - 0045-7825
M1 - 118034
ER -