scholarly journals Discontinuous Galerkin Isogeometric Analysis of Elliptic PDEs on Surfaces

2016 ◽  
pp. 319-326 ◽  
Author(s):  
Ulrich Langer ◽  
Stephen E. Moore
Keyword(s):  
Discontinuous Galerkin ◽  
Isogeometric Analysis ◽  
Elliptic Pdes

A time discontinuous Galerkin isogeometric analysis method for non-Fourier thermal wave propagation problem

2019 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Yang Xia ◽  
Pan Guo
Keyword(s):  
Wave Propagation ◽  
Discontinuous Galerkin ◽  
Heat Wave ◽  
Isogeometric Analysis ◽  
Numerical Instability ◽  
Analysis Method ◽  
Content Type ◽  
Mesh Dependency ◽  
Spurious Oscillations ◽  
Stability And Accuracy

Purpose Numerical instability such as spurious oscillation is an important problem in the simulation of heat wave propagation. The purpose of this study is to propose a time discontinuous Galerkin isogeometric analysis method to reduce numerical instability of heat wave propagation in the medium subjected to heat sources, particularly heat impulse. Design/methodology/approach The essential vectors of temperature and the temporal gradients are assumed to be discontinuous and interpolated individually in the discretized time domain. The isogeometric analysis method is applied to use its property of smooth description of the geometry and to eliminate the mesh-dependency. An artificial damping scheme with proportional stiffness matrix is brought into the final discretized form to reduce the numerical spurious oscillations. Findings The numerical spurious oscillations in the simulation of heat wave propagation are effectively eliminated. The smooth description of geometry with spline functions solves the mesh-dependency problem and improves the numerical precision. Originality/value The time discontinuous Galerkin method is applied within the isogeometric analysis framework. The proposed method is effective in the simulation of the wave propagation problems subjecting to impulse load with numerical stability and accuracy.

Interior Penalty Discontinuous Galerkin Based Isogeometric Analysis for Allen-Cahn Equations on Surfaces

2015 ◽  
Vol 18 (5) ◽  
pp. 1380-1416 ◽  
Author(s):  
Futao Zhang ◽  
Yan Xu ◽  
Falai Chen ◽  
Ruihan Guo
Keyword(s):  
Error Estimate ◽  
Discontinuous Galerkin ◽  
Isogeometric Analysis ◽  
Approximation Error ◽  
Generation Process ◽  
Element Analysis ◽  
Discrete Energy ◽  
Interior Penalty ◽  
Multiple Patches ◽  
3D Surfaces

AbstractWe propose a method that combines Isogeometric Analysis (IGA) with the interior penalty discontinuous Galerkin (IPDG) method for solving the Allen-Cahn equation, arising from phase transition in materials science, on three-dimensional (3D) surfaces consisting of multiple patches. DG ideology is adopted at patch level, i.e., we employ the standard IGA within each patch, and employ the IPDG method across the patch interfaces. IGA is very suitable for solving Partial Differential Equations (PDEs) on (3D) surfaces and the IPDG method is used to glue the multiple patches together to get the right solution. Our method takes advantage of both IGA and the IPDG method, which allows us to design a superior semi-discrete (in time) IPDG scheme. First and most importantly, the time-consuming mesh generation process in traditional Finite Element Analysis (FEA) is no longer necessary and refinements, includingh-refinement andp-refinement which both maintain the original geometry, can be easily performed at any level. Moreover, the flexibility of the IPDG method makes our method very easy to handle cases with non-conforming patches and different degrees across the patch interfaces. Additionally, the geometrical error is eliminated (for all conic sections) or significantly reduced at the beginning due to the geometric flexibility of IGA basis functions, especially the use of multiple patches. Finally, this method can be easily formulated and implemented. We present our semi-discrete IPDG scheme after generally describe the problem, and then briefly introduce the time marching method employed in this paper. Theoretical analysis is carried out to show that our method satisfies a discrete energy law, and achieves the optimal convergence rate with respect to theL2norm. Furthermore, we propose an elliptic projection operator on (3D) surfaces and prove an approximation error estimate which are vital for us to obtain the error estimate in theL2norm. Numerical tests are given to validate the theory and gauge the good performance of our method.

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