We reconsider a linear finite element method on a Bakhvalov-type mesh for a singularly perturbed problem with two parameters , which has been discussed in Brdar and Zarin (J. Comput. Appl. Math., 292 (2016), pp. 307--319). A novel interpolant is introduced for convergence analysis. From this interpolation, error analysis with Lagrange interpolation on the Bakhvalov-type mesh becomes feasible. As a result, we not only prove an optimal convergence between the true solution and the numerical solution, but also get a supercloseness result between the Lagrange interpolant and the numerical solution. More importantly, our arguments can be easily generalized to more general situations .
In this paper, the linear finite element method on a Bakhvalov-type mesh is applied to a singularly perturbed problem with two parameters . The solution of the problem exists two exponential boundary layers. A new interpolation, which is simple in construction and analysis, is introduced for convergence analysis. Furthermore, we find a subtle relationship between the Bakhvalov-type mesh itself and the weaker exponential layer and obtain an interesting result. Finally, we prove a supercloseness result between the Lagrange interpolation and the numerical solution. Numerical tests confirm our theoretical results .
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We reconsider a
In this paper, the
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for
is applied to
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, which has been discussed in Brdar and Zarin (J. Comput. Appl. Math., 292 (2016), pp. 307--319). A novel interpolant
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From this interpolation, error analysis with Lagrange interpolation on
Furthermore, we find a subtle relationship between
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becomes feasible. As a result, we not only prove an optimal convergence between the true solution and the numerical solution, but also get
itself and the weaker exponential layer and obtain an interesting result. Finally, we prove
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interpolant
interpolation
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More importantly, our arguments can be easily generalized to more general situations
Numerical tests confirm our theoretical results