Method for constructing meshes adapting to the solution of boundary value problems for ordinary differential equations of the second and fourth orders

To solve boundary value problems for ordinary differential equations of the second and fourth orders, we suggest a method for constructing a sequence of adaptively refined and coarsened meshes in a version of the finite element method with piecewise cubic Hermite basis functions. The construction of...

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Bibliographic Details
Published in:Differential equations 2009-08, Vol.45 (8), p.1189-1202
Main Authors: Zolotareva, N. D., Nikolaev, E. S.
Format: Article
Language:English
Subjects:
Boundary value problems
Difference and Functional Equations
Differential equations
Mathematics
Mathematics and Statistics
Methods
Numerical Methods
Ordinary Differential Equations
Partial Differential Equations
Studies
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Summary:To solve boundary value problems for ordinary differential equations of the second and fourth orders, we suggest a method for constructing a sequence of adaptively refined and coarsened meshes in a version of the finite element method with piecewise cubic Hermite basis functions. The construction of the meshes is based on C -norm estimates of the variation in the approximate solution or in the value of the functional to be minimized under the addition of a test point to a mesh interval or the deletion of a point from the current mesh. We present the results of numerical experiments used to assess the efficiency of the method. By way of example, problems whose solutions have singularities of the boundary layer type were used in these experiments. We carry out a comparison with a version of the method based on the uniform mesh refinement.
ISSN:0012-2661
1608-3083
DOI:10.1134/S0012266109080102