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1.
2.
We discuss an error estimation procedure for the global error of collocation schemes applied to solve singular boundary value problems with a singularity of the first kind. This a posteriori estimate of the global error was proposed by Stetter in 1978 and is based on the idea of Defect Correction, originally due to Zadunaisky. Here, we present a new, carefully designed modification of this error estimate which not only results in less computational work but also appears to perform satisfactorily for singular problems. We give a full analytical justification for the asymptotical correctness of the error estimate when it is applied to a general nonlinear regular problem. For the singular case, we are presently only able to provide computational evidence for the full convergence order, the related analysis is still work in progress. This global estimate is the basis for a grid selection routine in which the grid is modified with the aim to equidistribute the global error. This procedure yields meshes suitable for an efficient numerical solution. Most importantly, we observe that the grid is refined in a way reflecting only the behavior of the solution and remains unaffected by the unsmooth direction field close to the singular point. 相似文献
3.
Volker Elling. 《Mathematics of Computation》2006,75(256):1721-1733
A particular case of initial data for the two-dimensional Euler equations is studied numerically. The results show that the Godunov method does not always converge to the physical solution, at least not on feasible grids. Moreover, they suggest that entropy solutions (in the weak entropy inequality sense) are not well posed.
4.
A FUNDAMENTAL SOLUTION FOR THE LAPLACE OPERATOR ON THE QUATERNIONIC HEISENBERG GROUP 总被引:2,自引:0,他引:2
朱理 《数学物理学报(B辑英文版)》2002,22(3)
In this paper, the author studies the Laplace operator on the quaternionic Heisenberg group, construct a fundamental solution for it and use this solution to prove the Lp-boundedness and the weak (1-1) boundedness of certain singular convolution operators on the quaternionic Heisenberg group. 相似文献
5.
Milan Medved? 《Journal of Mathematical Analysis and Applications》2002,267(2):643-650
A criterion for the nonexplosion of solutions to semilinear evolution equations on Banach spaces is proved. The result is obtained by applying a modification of the Bihari type inequality to the case of a weakly singular nonlinear integral inequality. 相似文献
6.
非线性奇异边值问题的正解 总被引:1,自引:0,他引:1
路慧芹 《应用泛函分析学报》2002,4(3):217-221
利用锥映射的不动点指数定量,研究了一类非线性奇异边值问题多个正解的存在性问题。在构造的解的存在条件之下,证明了奇异边值问题至少有两个正解的存在性定理。 相似文献
7.
Mingxin Wang 《Journal of Mathematical Analysis and Applications》2002,274(1):424-436
This paper deals with positive solutions of degenerate and quasilinear parabolic systems not in divergence form: ut=up(Δu+av), vt=vq(Δv+bu), with null Dirichlet boundary conditions and positive initial conditions, where p, q, a and b are all positive constants. The local existence and uniqueness of classical solution are proved. Moreover, it will be proved that all solutions exist globally if and only if ab?λ12, where λ1 is the first eigenvalue of −Δ in Ω with homogeneous Dirichlet boundary condition. 相似文献
8.
Deborah M. Greaves 《国际流体数值方法杂志》2006,50(6):693-711
An adaptive hierarchical grid‐based method for predicting complex free surface flows is used to simulate collapse of a water column. Adapting quadtree grids are combined with a high‐resolution interface‐capturing approach and pressure‐based coupling of the Navier–Stokes equations. The Navier–Stokes flow solution scheme is verified for simulation of flow in a lid‐driven cavity at Re=1000. Two approaches to the coupling of the Navier–Stokes equations are investigated as are alternative face velocity and hanging node interpolations. Collapse of a water column as well as collapse of a water column and its subsequent interaction with an obstacle are simulated. The calculations are made on uniform and adapting quadtree grids, and the accuracy of the quadtree calculations is shown to be the same as those made on the equivalent uniform grids. Results are in excellent agreement with experimental and other numerical data. A sharp interface is maintained at the free surface. The new adapting quadtree‐based method achieves a considerable saving in the size of the computational grid and CPU time in comparison with calculations made on equivalent uniform grids. Copyright © 2005 John Wiley & Sons, Ltd. 相似文献
9.
10.
Existence, Multiplicity and Infinite Solvability of Positive Solutions for One-Dimensional p-Laplacian 总被引:5,自引:0,他引:5
Qing Liu YAO 《数学学报(英文版)》2005,21(4):691-698
The existence, multiplicity and infinite solvability of positive solutions are established for some two-point boundary value problems of one-dimensional p-Laplacian. In this paper, by multiplicity we mean the existence of m solutions, where m is an arbitrary natural number. 相似文献