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This paper is concerned with finite element approximations of $W^{2,p}$strong solutions of second-order linear elliptic partial differential equations(PDEs) in non-divergence form with continuous coefficients. A nonstandard(primal) finite element method, which uses finite-dimensional subspacesconsisting globally continuous piecewise polynomial functions, is proposed andanalyzed. The main novelty of the finite element method is to introduce aninterior penalty term, which penalizes the jump of the flux across the interiorelement edges/faces, to augment a nonsymmetric piecewise defined andPDE-induced bilinear form. Existence, uniqueness and error estimate in adiscrete $W^{2,p}$ energy norm are proved for the proposed finite elementmethod. This is achieved by establishing a discrete Calderon-Zygmund-typeestimate and mimicking strong solution PDE techniques at the discrete level.Numerical experiments are provided to test the performance of proposed finiteelement method and to validate the convergence theory.

$C^0$ discontinuous Galerkin finite element methods for second order linear elliptic partial differential equations in non-divergence form