An extended finite element method with arbitrary interior discontinuous gradients is applied to two-phase immiscible flow problems. The discontinuity in the derivative of the velocity field is introduced by an enrichment with an extended basis whose gradient is discontinuous across the interface. Therefore, the finite element approximation can capture the discontinuities at the interface without requiring the mesh to conform to the interface, eliminating the need for remeshing. The equations for incompressible flow are solved by a fractional step method where the advection terms are stabilized by a characteristic Galerkin method. The phase interfaces are tracked by level set functions which are discretized by the same finite element mesh and are updated via a stabilized conservation law. The method is demonstrated in several examples.

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