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The mesh is largely concentrated in these regions

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Unformatted text preview: s occur. The initial mesh consisted of 28,437 elements. This rose to more than 400,000 elements during the adaptive enrichment. This computation was done on 16 processors of a parallel computer. The coloring of the images on the right of Figure 10.3 indicates processor assignments. The discontinuous Galerkin method is still evolving and many questions regarding ux evaluation, limiting, a posteriori error estimation, the treatment of di usive problems, and its e ciency relative to standard nite element methods remain unanswered. Problems 1. Construct a typical term in the mass matrix on the canonical element by integrating Z 1 Z 1; Nm ( )Nn( )d d 0 0 using the basis of monomials (10.3.5). 2. Use the monomial basis (10.3.5) and the Gram-Schmidt process of Figure 10.3.2 to construct an orthogonal basis on the canonical right triangle for polynomials of 40 Hyperbolic Problems Figure 10.3.6: Mach contours (left) and adaptive meshes (right) used to solve the compressible ow problem of Example 10.3.2 at t = 0 (top) and t = 10:1 (bottom). degree p = 2 or less. Bibliography 1] M. Abromowitz and I.A. Stegun. Handbook of Mathematical Functions, volume 55 of Applied Mathematics Series. National Bureau of Standards, Gathersburg, 1964. 2] S. Adjerid, K.D. Devine, J.E. Flaherty, and L. Krivodonova. A posteriori error estimation for discontinuous Galerkin solutions of hyperbolic problems. In preparation, 2000. 3] F. Bassi and S. Rebay. A high-order accurate discontinuous nite element method for the numerical solution of the compressible navier-stokes equations. Journal of Computational Physics, 131:267{279, 1997. 4] C.E. Baumann and J.T. Oden. A discontinuous hp nite element method for convection-di usion problems. to appear, 1999. 5] K.S. Bey and J.T. Oden. hp-version discontinuous galerkin method for hyperbolic conservation laws. Computer Methods in Applied Mechanics and Engineering, 133:259{286, 1996. 6] K.S. Bey, J.T. Oden, and A. Patra. hp-version discontinuous galerkin method for hyperb...
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