A Hybridized Discontinuous Galerkin Method for A Linear Degenerate Elliptic Equation Arising from Two-Phase Mixtures
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We develop a high-order hybridized discontinuous Galerkin (HDG) method for a linear degenerate elliptic equation arising from a two-phase mixture of mantle convection or glacier dynamics. We show that the proposed HDG method is well-posed by using an energy approach. We derive a priori error estimates for the method on simplicial meshes in both two- and three-dimensions. The error analysis shows that the convergence rates are optimal for both the scaled pressure and the scaled velocity for non-degenerate problems and are sub-optimal by half order for degenerate ones. Several numerical results are presented to confirm the theoretical estimates. Degenerate problems with low regularity solutions are also studied, and numerical results show that high-order methods are beneficial in terms of accuracy.
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