On the monotonicity and discrete maximum principle of the finite difference implementation of $$C^0$$-$$Q^2$$ finite element method

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We show that the fourth order accurate finite difference implementation of continuous finite element method with tensor product of quadratic polynomial basis is monotone thus satisfies the discrete maximum principle for solving a scalar variable coefficient equation $$-\nabla \cdot (a\nabla u)+cu=f$$ under a suitable mesh constraint.