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The $ K$-operator and the Galerkin method for strongly elliptic equations on smooth curves: local estimates


Author: Thanh Tran
Journal: Math. Comp. 64 (1995), 501-513
MSC: Primary 65R20; Secondary 41A15, 65N38
DOI: https://doi.org/10.1090/S0025-5718-1995-1284671-2
MathSciNet review: 1284671
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Abstract: Superconvergence in the $ {L^2}$-norm for the Galerkin approximation of the integral equation $ Lu = f$ is studied, where L is a strongly elliptic pseudodifferential operator on a smooth, closed or open curve. Let $ {u_h}$ be the Galerkin approximation to u. By using the K-operator, an operator that averages the values of $ {u_h}$, we will construct a better approximation than $ {u_h}$ itself. That better approximation is a legacy of the highest order of convergence in negative norms. For Symm's equation on a slit the same order of convergence can be recovered if the mesh is suitably graded.


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DOI: https://doi.org/10.1090/S0025-5718-1995-1284671-2
Article copyright: © Copyright 1995 American Mathematical Society