Finite volume schemes for the biharmonic problem on general meshes

Abstract

During the development of a convergence theory for Nicolaides' extension [21,24] of the classical MAC scheme [25,22,26] for the incompressible Navier-Stokes equations to unstructured triangle meshes, it became clear that a convergence theory for a new kind of finite volume discretizations for the biharmonic problem would be a very useful tool in the convergence analysis of the generalized MAC scheme. Therefore, we present and analyze new finite volume schemes for the approximation of a biharmonic problem with Dirichlet boundary conditions on grids which satisfy an orthogonality condition. We prove that a piece-wise constant approximate solution of the biharmonic problem converges in L 2 (Ω) to the exact solution. Similar results are shown for the discrete approximate of the gradient and the discrete approximate of the Laplacian of the exact solution. Error estimates are also derived. This part of the paper is a first, significant step towards a convergence theory of Nicolaides' extension of the classical MAC scheme. Further, we show that finite volume discretizations for the biharmonic problem can also be defined on very general, nonconforming meshes, such that the same convergence results hold. The possibility to construct a converging lowest order finite volume method for the H 2 -regular biharmonic problem on general meshes seems to be an interesting result for itself and clarifies the necessary ingredients for converging discretizations of the biharmonic problem. All these results are confirmed by numerical results.

Approximation of the biharmonic problem using P1 finite elements

“…The total number of triangles is then 2 N 2 raf , and the size of the mesh is of order 1/N raf . In the figure below, we show one of the meshes used, and in the table below, we present the results obtained using the scheme (10 In this 2D case, we again observe that the order of convergence is much better than 1/5. Turning to less regular meshes, we consider the case where the simplicial meshes are generated by the repetition of the same square pattern.…”

“…We then remark that, for all v ∈ V T ,0 , we have (2); let T be a conforming simplicial mesh of Ω and u T ∈ V T ,0 be the solution of (10). Then, as h T tends to 0 with θ ≤ θ T , for a fixed value of θ > 0:…”

“…All the above methods are high order methods, and therefore, rather computationally expensive and may not be so easy to implement. Recently, a cheaper low order method based on the discretization of the Laplace operator by a cell centred finite volume scheme was proposed [10]. The idea developed in the present paper is to use the discretization of the Laplace operator, which is naturally provided by the continuous piecewise linear finite element method (see Section 2).…”

On MAC schemes on triangular delaunay meshes, their convergence and application to coupled flow problems

“…Proof. We consider the discrete stream function v defined by Lemma 3.3 of [17] for f = rot w and define (I M w)| σ := curl σ v. Equation (31) of [17] gives (3.12). Estimate (3.11) is obtained using the result on the strong discrete reconstruction of the gradient of the discrete stream function, provided in [21].…”

Approximation of the biharmonic problem using piecewise linear finite elements

“…All the above methods are high order methods, and therefore, rather computationally expensive and may not be so easy to implement. Recently, a cheaper low order method based on the discretization of the Laplace operator by a cell centred finite volume scheme was proposed [9]. The idea in the present paper is to show that the discretization of the Laplace operator by the piecewise linear finite element method is also possible in order to obtain a convergent low order scheme for the biharmonic operator.…”