Convergence Analysis and Cost Estimate of an MLMC-HDG Method for Elliptic PDEs with Random Coefficients
Abstract
:1. Introduction
2. Preliminaries
2.1. Problem Setting
2.2. HDG Method
3. Error Analysis
3.1. HDG Projection
3.2. Flux Estimate
3.3. Pressure Estimate
4. Multilevel Monte Carlo HDG Method
4.1. The MC Method
4.2. The MLMC Method
5. Numerical Experiments
6. Discussion
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
- Holden, H.; Oksendal, B.; Uboe, J.; Zhang, T. Stochastic Partial Differential Equations—A Modeling, White Noise Functional Approach, 2nd ed.; Springer: New York, NY, USA, 2010; pp. 213–257. [Google Scholar]
- Ragusa, M.A. Regularity of solutions of divergence form elliptic equations. Proc. Am. Math. Soc. 2000, 128, 533–540. [Google Scholar] [CrossRef]
- Roberts, J.B.; Spanos, P.D. Random Vibration and Statistical Linearization; Dover Pubilications: New York, NY, USA, 2003; pp. 1–57. [Google Scholar]
- Zakaria, A. Stochastic system for generalized polytropic filtration. Math. Appl. Sci. 2020, 43, 134–173. [Google Scholar] [CrossRef]
- Babusˇka, I.; Nobile, F.; Tempone, R. A stochastic collocation method for elliptic partial differential equations with random input data. SIAM Rev. 2010, 52, 317–335. [Google Scholar] [CrossRef] [Green Version]
- Babusˇka, I.; Tempone, R.; Zouraris, G. Galerkin finite element approximations of stochastic elliptic partial differential equations. SIAM J. Numer. Anal. 2004, 42, 800–825. [Google Scholar] [CrossRef]
- Barth, A.; Schwab, C.; Zollinger, N. Multi-level Monte Carlo finite element method for elliptic PDEs with stochastic coefficients. Numer. Math. 2011, 119, 123–161. [Google Scholar] [CrossRef] [Green Version]
- Bespalov, A.; Powell, C.E.; Silvester, D. A priori error analysis of stochastic Galerkin mixed approximations of elliptic PDEs with random data. SIAM J. Numer. Anal. 2012, 50, 2039–2063. [Google Scholar] [CrossRef] [Green Version]
- Lord, G.J.; Powell, C.E.; Shardlow, T. An Introduction to Computational Stochastic PDEs; Cambridge University Press: New York, NY, USA, 2014; pp. 396–421. [Google Scholar]
- Caflish, R.E. Monte Carlo and quasi-Monte Carlo methods. Acta Numer. 2007, 7, 1–49. [Google Scholar] [CrossRef] [Green Version]
- Giles, M.B. Multilevel Monte Carlo methods. Acta Numer. 2015, 24, 259–328. [Google Scholar] [CrossRef] [Green Version]
- Gunzburger, M.D.; Webster, C.G.; Zhang, G. Stochastic finite element methods for partial differential equations with random input data. Acta Numer. 2014, 23, 521–650. [Google Scholar] [CrossRef]
- Koley, U.; Ray, D.; Sarkar, T. Multilevel Monte Carlo Finite Difference Methods for Fractional Conservation Laws with Random Data. SIAM/ASA J. Uncertain. Quantif. 2021, 9, 65–105. [Google Scholar] [CrossRef]
- Teckentrup, A.L.; Scheichl, R.; Giles, M.B.; Ullmann, E. Further analysis of multilevel Monte Carlo methods for elliptic PDEs with random coefficients. Numer. Math. 2013, 125, 569–600. [Google Scholar] [CrossRef] [Green Version]
- Cockburn, B.; Gopalakrishnan, J.; Lazarov, R. Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems. SIAM J. Numer. Anal. 2009, 47, 1319–1365. [Google Scholar] [CrossRef]
- Cockburn, B.; Dong, B.; Guzma´n, J. A superconvergent LDG-hybridizable Galerkin method for second-order elliptic problems. Math. Comp. 2008, 77, 1887–1916. [Google Scholar] [CrossRef] [Green Version]
- Cockburn, B.; Gopalakrishnan, J.; Sayas, F.J. A projection-based error analysis of HDG methods. Math. Comput. 2010, 79, 1351–1367. [Google Scholar] [CrossRef] [Green Version]
- Chen, G.; Monk, P.B.; Zhang, Y.W. L∞ norm error estimates for HDG methods applied to the Poisson equation with an application to the Dirichlet boundary control problem. SIAM J. Numer. Anal. 2021, 59, 720–745. [Google Scholar] [CrossRef]
- Chen, G.; Pi, L.Y.; Xu, L.W.; Zhang, Y.W. A superconvergence ensemble HDG method for parameterized convection diffusion equations. SIAM J. Numer. Anal. 2019, 57, 2551–2578. [Google Scholar] [CrossRef] [Green Version]
- Cockburn, B.; Singler, J.R.; Zhang, Y. Interpolatory HDG Method for Parabolic Semilinear PDEs. J. Sci. Comput. 2019, 79, 1777–1800. [Google Scholar] [CrossRef] [Green Version]
- Costa-Sole´, A.; Ruiz-Girones, E.; Sarrate, J. High-Order Hybridizable Discontinuous Galerkin Formulation for One-Phase Flow Through Porous Media. J. Sci. Comput. 2021, 87. [Google Scholar] [CrossRef]
- Yu, Y.; Chen, G.; Pi, L.Y.; Zhang, Y.W. A new ensemble HDG method for parameterized convection diffusion PDEs. Numer. Math. Theory Methods Appl. 2021, 14, 144–175. [Google Scholar] [CrossRef]
- Ciarlet, P. The Finite Element Method for Elliptic Problems; North-Holland: Amsterdam, The Netherlands, 1978; pp. 123–126. [Google Scholar]
- Brenner, S.C.; Scott, L.R. The Mathematical Theory of Finite Element Methods, 3rd ed.; Springer: Berlin, Germany, 2007; pp. 155–170. [Google Scholar]
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Li, M.; Luo, X. Convergence Analysis and Cost Estimate of an MLMC-HDG Method for Elliptic PDEs with Random Coefficients. Mathematics 2021, 9, 1072. https://doi.org/10.3390/math9091072
Li M, Luo X. Convergence Analysis and Cost Estimate of an MLMC-HDG Method for Elliptic PDEs with Random Coefficients. Mathematics. 2021; 9(9):1072. https://doi.org/10.3390/math9091072
Chicago/Turabian StyleLi, Meng, and Xianbing Luo. 2021. "Convergence Analysis and Cost Estimate of an MLMC-HDG Method for Elliptic PDEs with Random Coefficients" Mathematics 9, no. 9: 1072. https://doi.org/10.3390/math9091072