scholarly journals Discontinuous Galerkin Isogeometric Analysis of Convection Problem on Surface

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 497
Author(s):  
Liang Wang ◽  
Chunguang Xiong ◽  
Xinpeng Yuan ◽  
Huibin Wu
Keyword(s):  
Finite Element ◽  
Discontinuous Galerkin ◽  
Finite Element Methods ◽  
Isogeometric Analysis ◽  
Numerical Experiments ◽  
Numerical Scheme ◽  
Convergence Order ◽  
Optimal Convergence ◽  
Convection Problem ◽  
Implicitly Defined Surfaces

The objective of this work is to study finite element methods for approximating the solution of convection equations on surfaces embedded in R3. We propose the discontinuous Galerkin (DG) isogeometric analysis(IgA) formulation to solve convection problems on implicitly defined surfaces. Three numerical experiments shows that the numerical scheme converges with the optimal convergence order.

scholarly journals Analysis of the SBP-SAT Stabilization for Finite Element Methods Part I: Linear Problems

2020 ◽  
Vol 85 (2) ◽  
Author(s):  
R. Abgrall ◽  
J. Nordström ◽  
P. Öffner ◽  
S. Tokareva
Keyword(s):  
Finite Element ◽  
Boundary Conditions ◽  
Discontinuous Galerkin ◽  
Finite Element Methods ◽  
Main Idea ◽  
Galerkin Methods ◽  
Exact Integration ◽  
Galerkin Scheme ◽  
Continuous Galerkin ◽  
Continuous Galerkin Methods

AbstractIn the hyperbolic community, discontinuous Galerkin (DG) approaches are mainly applied when finite element methods are considered. As the name suggested, the DG framework allows a discontinuity at the element interfaces, which seems for many researchers a favorable property in case of hyperbolic balance laws. On the contrary, continuous Galerkin methods appear to be unsuitable for hyperbolic problems and there exists still the perception that continuous Galerkin methods are notoriously unstable. To remedy this issue, stabilization terms are usually added and various formulations can be found in the literature. However, this perception is not true and the stabilization terms are unnecessary, in general. In this paper, we deal with this problem, but present a different approach. We use the boundary conditions to stabilize the scheme following a procedure that are frequently used in the finite difference community. Here, the main idea is to impose the boundary conditions weakly and specific boundary operators are constructed such that they guarantee stability. This approach has already been used in the discontinuous Galerkin framework, but here we apply it with a continuous Galerkin scheme. No internal dissipation is needed even if unstructured grids are used. Further, we point out that we do not need exact integration, it suffices if the quadrature rule and the norm in the differential operator are the same, such that the summation-by-parts property is fulfilled meaning that a discrete Gauss Theorem is valid. This contradicts the perception in the hyperbolic community that stability issues for pure Galerkin scheme exist. In numerical simulations, we verify our theoretical analysis.

scholarly journals Multigrid solvers for immersed finite element methods and immersed isogeometric analysis

2019 ◽  
Vol 65 (3) ◽  
pp. 807-838 ◽  
Author(s):  
F. de Prenter ◽  
C. V. Verhoosel ◽  
E. H. van Brummelen ◽  
J. A. Evans ◽  
C. Messe ◽  
...  
Keyword(s):  
Finite Element ◽  
Finite Element Methods ◽  
Isogeometric Analysis ◽  
Convergence Rates ◽  
Computational Cost ◽  
Iterative Solvers ◽  
System Matrix ◽  
B Splines ◽  
Immersed Finite Element ◽  
Immersed Finite Element Methods

AbstractIll-conditioning of the system matrix is a well-known complication in immersed finite element methods and trimmed isogeometric analysis. Elements with small intersections with the physical domain yield problematic eigenvalues in the system matrix, which generally degrades efficiency and robustness of iterative solvers. In this contribution we investigate the spectral properties of immersed finite element systems treated by Schwarz-type methods, to establish the suitability of these as smoothers in a multigrid method. Based on this investigation we develop a geometric multigrid preconditioner for immersed finite element methods, which provides mesh-independent and cut-element-independent convergence rates. This preconditioning technique is applicable to higher-order discretizations, and enables solving large-scale immersed systems at a computational cost that scales linearly with the number of degrees of freedom. The performance of the preconditioner is demonstrated for conventional Lagrange basis functions and for isogeometric discretizations with both uniform B-splines and locally refined approximations based on truncated hierarchical B-splines.

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