Design of an Essentially Non-Oscillatory Reconstruction Procedure on Finite-Element Type Meshes
INSTITUTE FOR COMPUTER APPLICATIONS IN SCIENCE AND ENGINEERING HAMPTON VA
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In this report, we have designed an essentially non-oscillatory reconstruction for functions defined on finite-element type meshes. Two related problems are studied the interpolation of possibly unsmooth multivariate functions on arbitrary meshes and the reconstruction of a function from its average in the control volumes surrounding the nodes of the mesh. Concerning the first problem, we have studied the behavior of the highest coefficients of the Lagrange interpolation function which may admit discontinuities of locally regular curves. This enables us to choose the best stencil for the interpolation. The Choice of the smallest possible number of stencils is addressed. Concerning the reconstruction problem, because of the very nature of the mesh, the only method that may work is the so called reconstruction via deconvolution method. Unfortunately, it is well suited only for regular meshes as we show, but we also show how to overcome this difficulty. The global method has the expected order of accuracy but is conservative up to a higher order quadrature formula only. Some numerical examples are given which demonstrate the efficiency of the method.
- Statistics and Probability