148 research outputs found

    Mollification in strongly Lipschitz domains with application to continuous and discrete De Rham complex

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    We construct mollification operators in strongly Lipschitz domains that do not invoke non-trivial extensions, are LpL^p stable for any real number p[1,]p\in[1,\infty], and commute with the differential operators \nabla, ×\nabla{\times}, and \nabla{\cdot}. We also construct mollification operators satisfying boundary conditions and use them to characterize the kernel of traces related to the tangential and normal trace of vector fields. We use the mollification operators to build projection operators onto general H1H^1-, H(curl)\mathbf{H}(\text{curl})- and H(div)\mathbf{H}(\text{div})-conforming finite element spaces, with and without homogeneous boundary conditions. These operators commute with the differential operators \nabla, ×\nabla{\times}, and \nabla{\cdot}, are LpL^p-stable, and have optimal approximation properties on smooth functions

    Numerical Simulations of Bouncing Jets

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    Bouncing jets are fascinating phenomenons occurring under certain conditions when a jet impinges on a free surface. This effect is observed when the fluid is Newtonian and the jet falls in a bath undergoing a solid motion. It occurs also for non-Newtonian fluids when the jets falls in a vessel at rest containing the same fluid. We investigate numerically the impact of the experimental setting and the rheological properties of the fluid on the onset of the bouncing phenomenon. Our investigations show that the occurrence of a thin lubricating layer of air separating the jet and the rest of the liquid is a key factor for the bouncing of the jet to happen. The numerical technique that is used consists of a projection method for the Navier-Stokes system coupled with a level set formulation for the representation of the interface. The space approximation is done with adaptive finite elements. Adaptive refinement is shown to be very important to capture the thin layer of air that is responsible for the bouncing

    Discontinuous Galerkin Methods for Friedrichs’ Systems. Part II. Second‐order Elliptic PDEs

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