A posteriori discontinuous finite element error estimation for two-dimensional hyperbolic problems

✍ Scribed by Slimane Adjerid; Thomas C. Massey

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 1011 KB
• Volume: 191
• Category: Article
• ISSN: 0045-7825
• DOI: 10.1016/s0045-7825(02)00502-9

✦ Synopsis

We analyze the discontinuous finite element errors associated with p-degree solutions for two-dimensional first-order hyperbolic problems. We show that the error on each element can be split into a dominant and less dominant component and that the leading part is Oðh pþ1 Þ and is spanned by two (p þ 1)-degree Radau polynomials in the x and y directions, respectively. We show that the p-degree discontinuous finite element solution is superconvergent at Radau points obtained as a tensor product of the roots of (p þ 1)-degree Radau polynomial. For a linear model problem, the pdegree discontinuous Galerkin solution flux exhibits a strong Oðh 2pþ2 Þ local superconvergence on average at the element outflow boundary. We further establish an Oðh 2pþ1 Þ global superconvergence for the solution flux at the outflow boundary of the domain. These results are used to construct simple, efficient and asymptotically correct a posteriori finite element error estimates for multi-dimensional first-order hyperbolic problems in regions where solutions are smooth.