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H^1-norm error estimate for a nonstandard finite element approximation of second-order linear elliptic PDEs in non-divergence form

09/30/2019
by   Xiaobing Feng, et al.
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This paper establishes the optimal H^1-norm error estimate for a nonstandard finite element method for approximating H^2 strong solutions of second order linear elliptic PDEs in non-divergence form with continuous coefficients. To circumvent the difficulty of lacking an effective duality argument for this class of PDEs, a new analysis technique is introduced; the crux of it is to establish an H^1-norm stability estimate for the finite element approximation operator which mimics a similar estimate for the underlying PDE operator recently established by the authors and its proof is based on a freezing coefficient technique and a topological argument. Moreover, both the H^1-norm stability and error estimate also hold for the linear finite element method.

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