Registered Data

Contents

1 [CT116]

[CT116]

Session Time & Room

CT116 (1/1) : 5C @E705 [Chair: DEBASISH PRADHAN]

Classification

CT116 (1/1) : Numerical methods for partial differential equations, boundary value problems (65N)

[00549] On the penalty approach in finite difference methods

Session Time & Room : 5C (Aug.25, 13:20-15:00) @E705
Type : Contributed Talk
Abstract : We introduce a finite difference method with the $H^1$ and $L^2$ penalties to solve the elliptic PDEs over curved complicated domains. The sharp convergence of the penalized solution to the original one is provided. The accuracy in both strategies is almost analogous, provided the penalty parameter $\epsilon$ is $O(h^2)$ in the $H^1$ penalty approach and $O(h^4)$ in the $L^2$ penalty approach. The iterative methods developed for the proposed idea are highly efficient and furnish the theoretical outcomes. Keywords: Finite difference method, Elliptic PDEs, Penalty, Curved domain, Cartesian mesh. References: 1. S. Kale, and D. Pradhan, Error estimates of fictitious domain method with an $H^1$ penalty approach for elliptic problems, Comp. Appl. Math., Vol. 41, (2022), pp. 1-21. 2.B. Maury, Numerical Analysis of a finite element/volume penalty method, SIAM J. Numer Anal., Vol. 47(2), pp. 1126-1148, (2009). 3.N. Saito and G. Zhou, Analysis of the fictitious domain method with an $L^2$-penalty for elliptic problems, Numer. Funct. Anal. Optim. Vol. 36, (2015), pp. 501-527. 4.H. Suito, and H. Kawarada, Numerical simulation of spilled oil by fictitious domain method, Japan J. Indust. Appl. Math., Vol. 21, (2004), pp. 219-236.
Classification : 65N85, 65N15
Format : Talk at Waseda University
Author(s) :
- DEBASISH PRADHAN (Defence Institute of Advanced Technology, Pune - 411025, India)
- Swapnil Kale (Defence Institute of Advanced Technology, Pune)

[00508] Imposing Neumann or Robin boundary conditions through a penalization method

Session Time & Room : 5C (Aug.25, 13:20-15:00) @E705
Type : Contributed Talk
Abstract : We will present an n-dimensional extension of a penalization method previously suggested for Neumann or Robin boundary conditions. The existence and uniqueness are obtained using Droniou's approach for non-coercive linear elliptic problems, and we develop a boundary layer approach to establish the convergence of the penalization method. We present two-dimensional numerical examples using adequate schemes suitable for advection dominated problems, and outline the application of this method to population dynamics subject to climate change.
Classification : 65N85, 65M85, 65N06, 65M06, 92D25
Format : Talk at Waseda University
Author(s) :
- Bouchra Bensiali (École Centrale Casablanca)
- Jacques Liandrat (École Centrale Marseille, I2M)

[01822] Domain decomposition for the Random Feature Method

Session Time & Room : 5C (Aug.25, 13:20-15:00) @E705
Type : Contributed Talk
Abstract : The random feature method (RFM) is a framework for solving PDEs sharing the merits of both traditional and machine learning-based algorithms. The direct method for optimization shows a high accuracy but faces acute memory and time-consuming issues with the increase of the scale of the problem. We introduced the domain decomposition into RFM and build a distributed, low-communication, and high-parallelism framework which relieves the pressure of storage and improves solving efficiency significantly in RFM.
Classification : 65N99, 65F20, 65-04, 65Y05
Format : Talk at Waseda University
Author(s) :
- Yifei Sun (Soochow University)
- Jingrun Chen (University of Science and Technology of China)
- Weinan E (Peking University)

[00860] Probabilistic Domain Decomposition: Challenging Amdahl's curse on partial differential equations.

Session Time & Room : 5C (Aug.25, 13:20-15:00) @E705
Type : Contributed Talk
Abstract : Probabilistic Domain Decomposition allows solving elliptic BVPs with remarkable scalability by taking advantage of probabilistic representations of BVPs. This representation is less convenient when dealing with non linear problems or even unknown in the case of the Helmholtz equation. However, these limitations can be circumvented by introducing some iterative schemes. In this presentation we aim to provide an insight on these algorithms alongside some proof of concept results obtained in FUGAKU and CINECA.
Classification : 65N75, 68W10, 65N55
Format : Online Talk on Zoom
Author(s) :
- Jorge Morón-Vidal (University Carlos III of Madrid)

[00547] Fictitious domain methods with finite elements and penalty over spread interface

Session Time & Room : 5C (Aug.25, 13:20-15:00) @E705
Type : Contributed Talk
Abstract : We present a spread interface approach in fictitious domain methods to decipher the elliptic PDEs depicted over curved complex domains. In this approach, we employ the $L^2$ penalty for a small tubular neighborhood $\Omega_{\delta}$ near $\partial\Omega$ in $\mathrm{R}\backslash\Omega$ in place of the substantial penalty for the whole fictitious part $\mathrm{R}\backslash\Omega$. We achieve strong convergence results concerning the penalty parameter $\epsilon$ in addition to the a priori estimates and stability analysis. We implement the linear finite elements and acquire the expected error estimates. The comprehensive numerical investigations support the mathematical findings, which also anticipate optimal convergence regardless of the convexity and shape of the domain. Keywords: Fictitious domain methods, Elliptic problems, Curved domain, Error estimates, Uniform mesh. References: 1. S. Kale, and D. Pradhan, An augmented interface approach in fictitious domain methods, Comput. Math. with Appl., Vol. 125, pp. 238-247, (2022). 2. B. Maury, Numerical Analysis of a finite element/volume penalty method, SIAM J. Numer Anal., Vol. 47(2), (2009), pp. 1126-1148. 3. N. Saito and G. Zhou, Analysis of the fictitious domain method with an $L^2$-penalty for elliptic problems, Numer. Funct. Anal. Optim., Vol. 36, (2015), pp. 501-527. 4. S. Zhang, Analysis of finite element domain embedding methods for curved domains using uniform grids, SIAM J. Numer. Anal., Vol. 46(6), (2008), pp. 2843-2866.
Classification : 65N85, 65N15, Numerical solutions to partial differential equations
Format : Online Talk on Zoom
Author(s) :
- Swapnil Kale (Defence Institute of Advanced Technology, Pune)
- Debasish Pradhan (Defence Institute of Advanced Technology, Pune)