Non Uniform Weighted Extended B-Spline Finite Element Analysis of Non Linear Elliptic Partial Differential Equations
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Non Uniform Weighted Extended B‑Spline Finite Element Analysis of Non Linear Elliptic Partial Differential Equations Ayan Chakraborty1 · B. V. Rathish Kumar2 Accepted: 27 October 2020 © Foundation for Scientific Research and Technological Innovation 2020
Abstract We propose a non uniform web spline based finite element analysis for elliptic partial differential equation with the gradient type nonlinearity in their principal coefficients like p-laplacian equation and Quasi-Newtonian fluid flow equations. We discuss the well-posednes of the problems and also derive the apriori error estimates for the proposed finite element analysis and obtain convergence rate of O(h𝛼 ) for 𝛼 > 0. Keywords Finite element · Non uniform web-spline · Error estimates Finite element method is one of the popular numerical techniques for solving partial differential equation modeling real life problems from science and engineering. Currently there is marked interest for meshless approach for solving boundary value problems as it significantly saves the cost and trouble of generating mesh, which are infinitesimal in many cases may turn out to be the computationally the most expensive job.Weighted extended B-splines is a finite element method (fem) in a infinitesimal costs mesh framework. The present work on nonlinear elliptic problems is based on non-uniform weighted extended B-splines (NUWEBS) fem which was originally proposed by H ö llig et al. [1–3] on trivial mesh framework. The p-laplacian equation used into the design of shock free airfoil and non-Newtonian fluid flow model used in understanding seepage through coarse grained porous media in some geological problems etc. have gradient type non linearity in their principal coefficients. They also occur in the description of non linear diffusion [4, 5] and filtration [6] , power law materials [7] and Quasi Newtonian flows [8]. Earlier in a grid based framework mixed finite element methods were developed and analyzed in [9, 10] for elliptic problems.Recently efforts have been made to solve non-linear fractional problems [11–14] analytically under certain assumptions and approximations leading to simplification of models to facilitate the analytical solution derivation. Such an approach may be of help in analytically solving the simplified version of the set of models under current * B. V. Rathish Kumar [email protected] Ayan Chakraborty [email protected] 1
IIT Kanpur, Kanpur, India
2
Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur, India
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Vol.:(0123456789)
Differential Equations and Dynamical Systems
consideration. Here, devoid of such simplifying assumptions, we are concerned with the finite element analysis in a grid less framework and provide the convergence analysis of weighted extended b-spline finite element analysis for p-laplacian equation and QuasiNewtonian flow model. An outline of the paper is as follows. We present some preliminary knowledge on the non-uniform weighted extended b-spline (WEB-S) space in “Non Uniform Weig
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