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The discontinuous Galerkin method for fractional degenerate convection-diffusion equations. (English) Zbl 1247.65128

A degenerate convection-diffusion equation with a fractional Lévy operator is investigated. This operator is known as the fractional Laplacian which represents a nonlocal generalization of the Laplace operator. Such nonlocal equations appear in different area of research, e.g., mathematical finance-option pricing models, in hydrodynamics and molecular biology.
The discontinuous Galerkin(DG) method plays an important tool for obtaining a reliable numerical solution for these problems namely the so-called local DG (LDG) or the direct DG (DDG) method. In the LDG method the convection-diffusion equation is rewritten as a first order system and then approximated by the DG method for conservation laws. The DDG method used as a direct application of the DG method to the convection-diffusion equation after a suitable numerical flux derived for the diffusion term.
In this paper both the LDG and the DDG method are investigated. \(L_2\) stability for both approximations are proved with various accuracies for linear and nonlinear cases. For convergence results in the nonlinear case an entropy solution for the continuous problem is used. Then the convergence of the DDG solution to the entropy solution for piecewise constant elements is derived. Numerical experiments illustrating the qualitative behavior of the solutions conclude the paper.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35K65 Degenerate parabolic equations
35K59 Quasilinear parabolic equations
35L67 Shocks and singularities for hyperbolic equations
35R11 Fractional partial differential equations
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