PRIMAL HYBRID FINITE ELEMENT METHOD FOR PARABOLIC PROBLEMS WITH GRADIENT TYPE NON-LINEARITY

doi:10.1007/s10473-026-0225-z

Acta mathematica scientia,Series B ›› 2026, Vol. 46 ›› Issue (2): 1011-1035.doi: 10.1007/s10473-026-0225-z

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PRIMAL HYBRID FINITE ELEMENT METHOD FOR PARABOLIC PROBLEMS WITH GRADIENT TYPE NON-LINEARITY

Ravina SHOKEEN¹, Ajit PATEL^2,*, Divay GARG³

1. Department of Applied Science and Humanities, World College of Technology and Management, Farukh Nagar, Gurgaon, Haryana 122506, India;
2. Department of Mathematics,The LNM Institute of Information Technology, Jaipur, Rajasthan 302031, India;
3. Department of Applied Science, Galgotias College of Engineering and Technology, Knowledge Park-II, Greater Noida, G.B. Nagar, U.P. 201310, India

Received:2023-11-20 Revised:2024-01-08 Published:2026-05-22
Contact: ^*Ajit PATEL, E-mail: apatel@lnmiit.ac.in
About author:Ravina SHOKEEN, E-mail: ravinashokeen@outlook.com; Divay GARG, E-mail: divaygarg2@gmail.com
Supported by:
The first two authors acknowledge the financial support provided by the DST-FIST program (Govt. of India) for setting up the computing lab facility called Center for Mathematical & Financial Computing (C-MFC) at the LNM Institute of Information Technology under the scheme "Fund for Improvement of Science and Technology" (FIST-No. SR/FST/MS-I/2018/24). The third author is supported by CSIR research fellowship.

Abstract

Abstract: This article develops the primal hybrid finite element method with Lagrange multipliers to approximate nonlinear parabolic initial-boundary value problems with gradient type non-linearity.A modified elliptic projection is used to produce optimal order error estimates for the semi-discrete and backward Euler-based complete discrete schemes. In addition, error estimates in $L^{\infty}$-norm are established which are optimal in nature. Superconvergence result of the gradient in $L^{\infty}$-norm is discussed for the error between the primal hybrid solution and elliptic projection. As a bi-product, the proposed analysis provides optimal error analysis for non-conforming CR-elements.Finally, numerical tests are performed to validate the theoretical findings.

Key words: primal hybrid method (PHM), modified elliptic projection, strongly non-linear parabolic problem, semi-discrete method, Lagrange multiplier, complete discrete method, numerical results

CLC Number:

65M60

Ravina SHOKEEN, Ajit PATEL, Divay GARG. PRIMAL HYBRID FINITE ELEMENT METHOD FOR PARABOLIC PROBLEMS WITH GRADIENT TYPE NON-LINEARITY[J].Acta mathematica scientia,Series B, 2026, 46(2): 1011-1035.

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PRIMAL HYBRID FINITE ELEMENT METHOD FOR PARABOLIC PROBLEMS WITH GRADIENT TYPE NON-LINEARITY

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