Superconvergence of bi-k Degree Time-Space Fully Discontinuous Finite Element for First-Order Hyperbolic Equations

2015 ◽  
Vol 7 (3) ◽  
pp. 323-337 ◽  
Author(s):  
Hongling Hu ◽  
Chuanmiao Chen
Keyword(s):  
Finite Element ◽  
Hyperbolic Equations ◽  
Optimal Convergence Rate ◽  
Galerkin Finite Element ◽  
First Order ◽  
Optimal Convergence ◽  
Time Space ◽  
Correction Technique ◽  
Discontinuous Galerkin Finite Element ◽  
Discontinuous Finite Element

AbstractIn this paper, we present a superconvergence result for the bi-k degree time-space fully discontinuous finite element of first-order hyperbolic problems. Based on the element orthogonality analysis (EOA), we first obtain the optimal convergence order of discontinuous Galerkin finite element solution. Then we use orthogonality correction technique to prove a superconvergence result at right Radau points, which is higher one order than the optimal convergence rate. Finally, numerical results are presented to illustrate the theoretical analysis.

Discontinuous Finite Element Phase-Field Modeling of Solidification Microstructure Formation

2004 ◽  
Author(s):  
Y. Shu ◽  
X. Ai ◽  
B. Q. Li
Keyword(s):  
Finite Element ◽  
Phase Field ◽  
Current Method ◽  
Great Promise ◽  
Adaptive Meshing ◽  
Galerkin Finite Element ◽  
Discontinuous Galerkin Finite Element ◽  
Discontinuous Finite Element ◽  
Discontinuous Model ◽  
Solidification Problem

A discontinuous Galerkin finite element computational methodology is presented for the solution of the coupled phase-field and heat conduction equations for modeling microstructure evolution during solidification. The details of the discontinuous formulation and the solution procedures are given. A major difference between the current method and those used in the literatures is the application of higher-order localized formulation and unstructured mesh, which holds a great promise in both parallel computing and adaptive meshing. The accuracy of the discontinuous model is checked with the analytic solution for a simple 1-D solidification problem. Numerical simulations and selected results are given for more complex 2-D dendrite structures formed during solidification. The calculated results are consistent with those reported in literature.

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