Volume 3, Issue 1
Numerical Solution of Non-Steady State Porous Flow Free Boundary Problems

Shao-Yun Huang & Cai-Jing Zhou

J. Comp. Math., 3 (1985), pp. 72-89

Published online: 1985-03

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  • Abstract

The aim of this paper is the study of the convergence of a finite elment approximation for a variational inequality related to free boundary problems in non-steady fluid flow through porous media. There have been many results in the stationary case, for example, the steady dam problems, the steady flow well problems, etc. In this paper we shall deal with the axisymmetric non-steady porous flow well problem. It is well know that by means of Torelli's transform this problem, similar to the non-steady rectangular dam problem, can be reduced a variational, inequality, and the existence, uniquenness and regularity of the solution can be obtained ([12, 7]). Now we study the numerical solution of this variational inequality. The main results are as follows: 1. We establish new regularity properties for the solution $W$ of the variation inequality. We prove that $W \in L^\infty(0, T; H^2(D))$, $\rho_0W\in L^\infty(0, T; H^2(T_n))$ and $D_1\rho_0W\in L^2(0, T; H^1(T_n))$ (see Theorem 2.5). Friedman and torelli [7] obtained $W\in L^2(0, T; H^2(D))$. Our new regularity properties will be used for error estimation. 2. We prove that the error estimate for the finite element solution of the variational inequality is $$ ( \sum^N_{i=1}\| W^1 - W^1_h \|^2_{H^1(D)}\Delta t)^{1/2} = O(h+\Delta t^{1/2})$$ (see Theorem 3.4). In the stationary case the error estimate is $\|W-W_h\|_{H^1(D)} = O(k)$ ([3,6]). 3. We give a numerical example and compare the result with the corresponding result in the stationary case. The result of this paper are valid for the non-ready rectangular dam problem with stationary or quasi-stationary initial data (see [7], p.534).

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@Article{JCM-3-72, author = {Shao-Yun Huang and Cai-Jing Zhou}, title = {Numerical Solution of Non-Steady State Porous Flow Free Boundary Problems}, journal = {Journal of Computational Mathematics}, year = {1985}, volume = {3}, number = {1}, pages = {72--89}, abstract = { The aim of this paper is the study of the convergence of a finite elment approximation for a variational inequality related to free boundary problems in non-steady fluid flow through porous media. There have been many results in the stationary case, for example, the steady dam problems, the steady flow well problems, etc. In this paper we shall deal with the axisymmetric non-steady porous flow well problem. It is well know that by means of Torelli's transform this problem, similar to the non-steady rectangular dam problem, can be reduced a variational, inequality, and the existence, uniquenness and regularity of the solution can be obtained ([12, 7]). Now we study the numerical solution of this variational inequality. The main results are as follows: 1. We establish new regularity properties for the solution $W$ of the variation inequality. We prove that $W \in L^\infty(0, T; H^2(D))$, $\rho_0W\in L^\infty(0, T; H^2(T_n))$ and $D_1\rho_0W\in L^2(0, T; H^1(T_n))$ (see Theorem 2.5). Friedman and torelli [7] obtained $W\in L^2(0, T; H^2(D))$. Our new regularity properties will be used for error estimation. 2. We prove that the error estimate for the finite element solution of the variational inequality is $$ ( \sum^N_{i=1}\| W^1 - W^1_h \|^2_{H^1(D)}\Delta t)^{1/2} = O(h+\Delta t^{1/2})$$ (see Theorem 3.4). In the stationary case the error estimate is $\|W-W_h\|_{H^1(D)} = O(k)$ ([3,6]). 3. We give a numerical example and compare the result with the corresponding result in the stationary case. The result of this paper are valid for the non-ready rectangular dam problem with stationary or quasi-stationary initial data (see [7], p.534). }, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/9608.html} }
TY - JOUR T1 - Numerical Solution of Non-Steady State Porous Flow Free Boundary Problems AU - Shao-Yun Huang & Cai-Jing Zhou JO - Journal of Computational Mathematics VL - 1 SP - 72 EP - 89 PY - 1985 DA - 1985/03 SN - 3 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/9608.html KW - AB - The aim of this paper is the study of the convergence of a finite elment approximation for a variational inequality related to free boundary problems in non-steady fluid flow through porous media. There have been many results in the stationary case, for example, the steady dam problems, the steady flow well problems, etc. In this paper we shall deal with the axisymmetric non-steady porous flow well problem. It is well know that by means of Torelli's transform this problem, similar to the non-steady rectangular dam problem, can be reduced a variational, inequality, and the existence, uniquenness and regularity of the solution can be obtained ([12, 7]). Now we study the numerical solution of this variational inequality. The main results are as follows: 1. We establish new regularity properties for the solution $W$ of the variation inequality. We prove that $W \in L^\infty(0, T; H^2(D))$, $\rho_0W\in L^\infty(0, T; H^2(T_n))$ and $D_1\rho_0W\in L^2(0, T; H^1(T_n))$ (see Theorem 2.5). Friedman and torelli [7] obtained $W\in L^2(0, T; H^2(D))$. Our new regularity properties will be used for error estimation. 2. We prove that the error estimate for the finite element solution of the variational inequality is $$ ( \sum^N_{i=1}\| W^1 - W^1_h \|^2_{H^1(D)}\Delta t)^{1/2} = O(h+\Delta t^{1/2})$$ (see Theorem 3.4). In the stationary case the error estimate is $\|W-W_h\|_{H^1(D)} = O(k)$ ([3,6]). 3. We give a numerical example and compare the result with the corresponding result in the stationary case. The result of this paper are valid for the non-ready rectangular dam problem with stationary or quasi-stationary initial data (see [7], p.534).
Shao-Yun Huang & Cai-Jing Zhou. (1970). Numerical Solution of Non-Steady State Porous Flow Free Boundary Problems. Journal of Computational Mathematics. 3 (1). 72-89. doi:
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