具有空间测量数据的抛物型方程界面问题的半离散误差估计
Semi-Discrete Error Estimation of the Interface Problem of Parabolic Equations with Spatial Measurement Data
DOI: 10.12677/ORF.2023.134412, PDF, 下载: 242  浏览: 311  国家自然科学基金支持
作者: 杨 勋, 罗贤兵：贵州大学，数学与统计学院，贵州 贵阳
关键词: 空间测度数据；半离散；L2范数；先验误差估计；Spatial Measure Data； Semi-Discrete； L2 Norm； A Priori Error Estimates
摘要: 本文研究了具有空间测度数据的线性抛物界面问题的先验误差分析，对于空间的离散我们运用有限元离散得到它的半离散问题，由于测度的正则性较低，所以问题的解在整个域内具有很低的正则性。 利用L2投影算子和对偶性参数，在最小正则性条件下，导出了空间离散有限元逼近的L2范数中的先验误差估计。
Abstract: In this paper, we study the a priori error analysis of linear parabolic interface problems with spatial measure data. For the spatial discretization, we use finite element discretization to obtain its semi-discrete problem; because the measure regularity is low, so the solution of the problem has very low regularity in the whole domain. We derive a prior error estimate in the L2 norm of the spatial discrete finite element approximation under minimum regularity, using the L2 projection operator and the duality parameter.
文章引用：杨勋, 罗贤兵. 具有空间测量数据的抛物型方程界面问题的半离散误差估计[J]. 运筹与模糊学, 2023, 13(4): 4120-4131. https://doi.org/10.12677/ORF.2023.134412

参考文献

[1] Shakya, P. and Sinha, R.K. (2019) A Posteriori Error Analysis for Finite Element Approxi- mations of Parabolic Optimal Control. Applied Numerical Mathematics, 136, 23-45.
https://doi.org/10.1016/j.apnum.2018.09.015
[2] Araya, R., Behrens, E. and Rodr´ıguez, R. (2006) A Posteriori Error Estimates for Elliptic Problems with Dirac Delta Source Terms. Numerische Mathematik, 105, 193-216.
https://doi.org/10.1007/s00211-006-0041-2
[3] Gong, W. (2013) Error Estimates for Finite Element Approximations of Parabolic Equations with Measure Data. Mathematics of Computation, 82, 69-98.
https://doi.org/10.1090/S0025-5718-2012-02630-5
[4] Casas, E. (2014) Optimal Control of Semilinear Elliptic Equations in Measure Spaces. SIAM Journal on Control and Optimization, 52, 339-564.
https://doi.org/10.1137/13092188X
[5] Shakya, P. and Sinha, R.K. (2021) Finite Element Approximations of Parabolic Optimal Con- trol Problem with Measure Data in Time. Applicable Analysis, 100, 2706-2734.
https://doi.org/10.1080/00036811.2019.1698722
[6] Chen, Z. and Zou, J. (1998) Finite Element Methods and Their Convergence for Elliptic and Parabolic Interface Problems. Numerische Mathematik, 79, 175-202.
https://doi.org/10.1007/s002110050336
[7] Huang, J. and Zou, J. (2002) Some New A Priori Estimates for Second-Order Elliptic and Parabolic Interface Problems. Journal of Differential Equations, 184, 570-586.
https://doi.org/10.1006/jdeq.2001.4154
[8] Lions, J.-L. and Magenes, E. (1972) Non-Homogeneous Boundary Value Problems and Applications. Vol. I, Die Grundlehren der mathematischen Wissenschaften, Springer Verlag, New York-Heidelberg. (Translated from the French by P. Kenneth)
[9] Bramble, J.H. and King, J.T. (1996) A Finite Element Method for Interface Problems in Domains with Smooth Boundaries and Interfaces. Advances in Computational Mathematics, 6, 109-138.
https://doi.org/10.1007/BF02127700
[10] Ciarlet, P.G. (2002) The Finite Element Method for Elliptic Problems. In: Classics in Applied Mathematics, Vol. 40, SIAM, Philadelphia.
https://doi.org/10.1137/1.9780898719208
[11] Rannacher, R. and Scott, R. (1982) Some Optimal Error Estimates for Piecewise Linear Finite Element Approximations. Mathematics of Computation, 38, 437-445.
https://doi.org/10.1090/S0025-5718-1982-0645661-4
[12] Thom´ee, V. (2006) Galerkin Finite Element Methods for Parabolic Problems. In: Springer Series in Computational Mathematics, 2nd Edition, Springer-Verlag, Berlin.
[13] Chrysafinos, K. and Hou, L.S. (2002) Error Estimates for Semidiscrete Finite Element Approximations of Linear and Semilinear Parabolic Equations under Minimal Regularity Assumptions.SIAM Journal on Numerical Analysis, 40, 282-306.
https://doi.org/10.1137/S0036142900377991
[14] Ciarlet, P.G. (1978) The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam.
https://doi.org/10.1115/1.3424474
[15] Sinha, R.K. and Deka, B. (2005) Optimal Error Estimates for Linear Parabolic Problems with Discontinuous Coefficients. SIAM Journal on Numerical Analysis, 43, 733-749.
https://doi.org/10.1137/040605357

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