New mixed finite element method on polygonal and polyhedral meshes.(English)Zbl 1048.65113
From the authors’ summary: In the framework of mixed and hybrid finite element methods for diffusion-type and other elliptic differential equations, it is necessary to apply numerical schemes with variables given as values of normal fluxes on the edges (faces) of the elementary cells and values of the scalar-valued function in each cell.
In this paper, we propose a general method of constructing finite element approximations on polygonal and polyhedral meshes, whose cells are convex and nonconvex polygonal domains in \(\mathbb{R}^2\) and polyhedrons in \(\mathbb{R}^3\). We present a natural way of constructing cell prolongation operators, which makes it possible to easily compute the coefficients of the respective mass matrices. Also, the proposed prolongations satisfy the important requirement that the image of the divergence operator on the extended fields belongs to the set of piecewise constant functions. The latter fact provides direct justification of the well-posedness of the arising discrete problems.
In this paper, we propose a general method of constructing finite element approximations on polygonal and polyhedral meshes, whose cells are convex and nonconvex polygonal domains in \(\mathbb{R}^2\) and polyhedrons in \(\mathbb{R}^3\). We present a natural way of constructing cell prolongation operators, which makes it possible to easily compute the coefficients of the respective mass matrices. Also, the proposed prolongations satisfy the important requirement that the image of the divergence operator on the extended fields belongs to the set of piecewise constant functions. The latter fact provides direct justification of the well-posedness of the arising discrete problems.
Reviewer: Xavier Antoine (Pasadena)
MSC:
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 35J25 | Boundary value problems for second-order elliptic equations |
| 65N50 | Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs |
Keywords:
mixed finite elements;diffusion-type equations;elliptic equations;polygonal and polyhedral meshesCite
Full Text:DOI
References:
| [1] | D. Braess. Finite Elements. Theory, Fast Solvers and Applications to Solid Mechanics. Cambridge University press, 1997. ·Zbl 0894.65054 |
| [2] | F. Brezzi and M. Fortin. Mixed and Hybrid Finite Element Methods. Springer, Berlin, 1991. ·Zbl 0788.73002 |
| [3] | J.E. Roberts and J.M. Thomas. Mixed and hybrid methods. In: Handbook of Numerical Analysis II, Finite Element Methods (Eds. P. Ciarlet and J. Lions). Elsevier/North Holland, Amsterdam, 1991, pp. 523-639. |
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