An a priori error estimate for a monotone mixed finite-element discretization of a convection–diffusion problem

Holst, Stefan

doi:10.1007/s00211-007-0097-7

Cited by 3 publications

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References 26 publications

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“…Much less is known for higher order methods such as spectral FEM, hp-FEM or finite volume methods [2,6,13,23]. Compared to the elliptic type, more restrictive conditions on mesh are required to obtain discrete maximum principle for the parabolic type equations, see [18,16,7,17].In this article, we study direct discontinuous Galerkin method [8] and its variations [9,19,21], and prove the polynomial solution satisfy discrete maximum principle with third order of accuracy. Discontinuous Galerkin (DG) method is a class of finite element method that use completely discontinuous piecewise functions as numerical approximations.…”

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Third order maximum-principle-satisfying direct discontinuous Galerkin methods for time dependent convection diffusion equations on unstructured triangular meshes

Chen

Huang

Yan

2016

Journal of Computational Physics

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We develop 3rd order maximum-principle-satisfying direct discontinuous Galerkin methods [8,9,19,21] for convection diffusion equations on unstructured triangular mesh. We carefully calculate the normal derivative numerical flux across element edges and prove that, with proper choice of parameter pair (β 0 , β 1 ) in the numerical flux, the quadratic polynomial solution satisfies strict maximum principle. The polynomial solution is bounded within the given range and third order accuracy is maintained. There is no geometric restriction on the meshes and obtuse triangles are allowed in the partition. A sequence of numerical examples are carried out to demonstrate the accuracy and capability of the maximum-principle-satisfying limiter. and practically cause the computations to blow up. Thus it is attractive to have the numerical solution satisfy discrete maximum principle (or preserve positivity). Solution of equation (1.1) may represent a specific physical meaning and is supposed to be positive, thus negative value approximation loses physical meanings in such cases.Generally it is very difficult to design high order numerical methods that satisfy discrete maximum principle for convection diffusion equations (1.1). No finite difference method is known to achieve better than second-order accuracy [5,25,20] that satisfies discrete maximum principle. Much less is known for higher order methods such as spectral FEM, hp-FEM or finite volume methods [2,6,13,23]. Compared to the elliptic type, more restrictive conditions on mesh are required to obtain discrete maximum principle for the parabolic type equations, see [18,16,7,17].In this article, we study direct discontinuous Galerkin method [8] and its variations [9,19,21], and prove the polynomial solution satisfy discrete maximum principle with third order of accuracy. Discontinuous Galerkin (DG) method is a class of finite element method that use completely discontinuous piecewise functions as numerical approximations. Since the basis functions can be completely discontinuous, these methods have the flexibility that is not shared by standard finite element methods, such as the allowance of arbitrary triangulations with hanging nodes, complete freedom of choosing polynomial degrees in each element (p adaptivity), and extremely local data structure and the resulting high parallel efficiency.Recently in [24,25,26], Zhang and Shu designed a maximum-principle-satisfying limiter for high order DG and finite volume methods for hyperbolic conservation laws. The key step in Zhang and Shu's discussion is to show the polynomial solution average falling in the given minimum and maximum bounds. For hyperbolic type equations, the solution average evolution only relies on the solution polynomial values on the element edges. For diffusion type equations, the evolution of solution average depends on the solution derivative values on the edges, thus the technique developed in [24] can not be applied.In [8], we developed the direct DG method (DDG) as a new diffusion solver. The key contrib...

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Third order maximum-principle-satisfying direct discontinuous Galerkin methods for time dependent convection diffusion equations on unstructured triangular meshes

Chen

Huang

Yan

2016

Journal of Computational Physics

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“…A comparison of the performance and efficiency of some of the various (not necessarily positivity preserving) numerical schemes for chemotaxis is presented in [38]. Among other numerous previous works are upwind finite element and central upwind finite volume methods [6,31,32], mixed finite elements [25] (that borrows techniques from similar semiconductor models [4,5,17]), discontinuous Galerkin (DG) methods [11,12], and a fractional step method [40]. The mixed and DG methods are particularly interesting since they are extendable to higher order (unlike our method, which is essentially first order).…”

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confidence: 99%