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A discontinuous finite volume method for the Stokes problems. (English) Zbl 1112.65125
The author extends the ideas developed in recent work to solve the Stokes equations on both triangular and rectangular meshes. In the author’s methods, the velocity is approximated by discontinuous piecewise linear functions on a triangular mesh and by discontinuous piecewise rotated bilinear functions on a rectangular mesh. Piecewise constant functions are used as test functions for the velocity in the discontinuous finite volume method. Therefore, after multiplying the differential equations by the test functions and integrating by parts, the area integrals in the formulations will disappear, which gives the simplicity of the finite volume method. One of the advantages of using discontinuous approximation functions is, that it is easy to build high order elements. Main result: Optimal error estimates for the velocity in the norm $\vert \vert \vert \cdot \vert \vert \vert$ and for the pressure in the $L_2$-norm are derived. An optimal error estimate for the approximation of the velocity in a mesh-dependent norm is obtained. Precise proofs of all theorems and lemmas are proposed.

##### MSC:
 65N30 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE) 65N15 Error bounds (BVP of PDE) 76D07 Stokes and related (Oseen, etc.) flows 35J25 Second order elliptic equations, boundary value problems 35Q30 Stokes and Navier-Stokes equations 76M12 Finite volume methods (fluid mechanics)
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