54 research outputs found
The discontinuous Galerkin method for fractional degenerate convection-diffusion equations
We propose and study discontinuous Galerkin methods for strongly degenerate
convection-diffusion equations perturbed by a fractional diffusion (L\'evy)
operator. We prove various stability estimates along with convergence results
toward properly defined (entropy) solutions of linear and nonlinear equations.
Finally, the qualitative behavior of solutions of such equations are
illustrated through numerical experiments
A high-order local discontinuous Galerkin method for the -Laplace equation
We study the high-order local discontinuous Galerkin (LDG) method for the
-Laplace equation. We reformulate our spatial discretization as an
equivalent convex minimization problem and use a preconditioned gradient
descent method as the nonlinear solver. For the first time, a weighted
preconditioner that provides -independent convergence is applied in the LDG
setting. For polynomial order , we rigorously establish the
solvability of our scheme and provide a priori error estimates in a
mesh-dependent energy norm. Our error estimates are under a different and
non-equivalent distance from existing LDG results. For arbitrarily high-order
polynomials under the assumption that the exact solution has enough regularity,
the error estimates demonstrate the potential for high-order accuracy. Our
numerical results exhibit the desired convergence speed facilitated by the
preconditioner, and we observe best convergence rates in gradient variables in
alignment with linear LDG, and optimal rates in the primal variable when .Comment: 36 pages, 36 figure
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