Development and analysis of higher order finite volume methods for elliptic equations

Moongyu Park, Purdue University

Abstract

Currently used finite volume methods are essentially low order methods. In this thesis, we present a systematic way to derive higher order finite volume schemes from higher order mixed finite element methods. Mostly for convenience but sometimes from necessity, our procedure starts from the hybridization of the mixed method. It then approximates the inner product of vector functions by an appropriate, critical quadrature rule; this allows the elimination of the flux and Lagrange multiplier parameters so as to obtain equations in the scalar variable, which will define the finite volume method. If we eliminate the flux and scalar variable, then we have the well-known Fraeijs de Veubeke methods [3],[20]. Following this derivation with different mixed finite element spaces leads to a variety of finite volume schemes. We begin by studying an efficient second-order finite volume method based on the Brezzi-Douglas-Fortin-Marini space (BDFM2) of index two. Then, we present a general global analysis of the difference between the solution of the underlying mixed finite element method and its related finite volume method. Then, we derive finite volume methods of all orders from the Raviart-Thomas two and three-dimensional rectangular elements; we also find finite volume methods to associate with BDFM 2 three-dimensional rectangles. We derive the Fraeijs de Veubeke method based on BDM1 triangular elements and BDFM1 prismatic elements. In each case, we obtain optimal error estimates for both the scalar and flux variable. At last section, we present computational results which confirm the optimal error estimates in §7 and show superconvergence of flux along Gauss lines in the sense of L2 and L∞ norms.

Degree

Ph.D.

Advisors

Douglas, Purdue University.

Subject Area

Mathematics

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