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We propose a Real-Space Gutzwiller variational approach and apply it to a system of repulsively interacting ultracold fermions with spin 1/2 trapped in an optical lattice with a harmonic confinement. Using the Real-Space Gutzwiller variational approach, we find that in system with balanced spin-mixtures on a square lattice, antiferromagnetism either appears in a checkerboard pattern or forms a ring and antiferromagnetic order is stable in the regions where the particle density is close to one, which is consistent with the recent results obtained by the Real-Space Dynamical Mean-field Theory approach. We also investigate the imbalanced case and find that antiferromagnetic order is suppressed there.
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