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This paper establishes the optimal $H^1$-norm error estimate for anonstandard finite element method for approximating $H^2$ strong solutions ofsecond order linear elliptic PDEs in non-divergence form with continuouscoefficients. To circumvent the difficulty of lacking an effective dualityargument for this class of PDEs, a new analysis technique is introduced; thecrux of it is to establish an $H^1$-norm stability estimate for the finiteelement approximation operator which mimics a similar estimate for theunderlying PDE operator recently established by the authors and its proof isbased on a freezing coefficient technique and a topological argument. Moreover,both the $H^1$-norm stability and error estimate also hold for the linearfinite element method.

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