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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 \(| | | \cdot | | | \) 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 element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
76D07 Stokes and related (Oseen, etc.) flows
35J25 Boundary value problems for second-order elliptic equations
35Q30 Navier-Stokes equations
76M12 Finite volume methods applied to problems in fluid mechanics
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