Efficient use of the local discontinuous Galerkin method for meshes sliding on a circular boundary (original) (raw)
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COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, 2001
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A conforming discontinuous Galerkin finite element method was introduced by Ye and Zhang, on simplicial meshes and on polytopal meshes, which has the flexibility of using discontinuous approximation and an ultra simple formulation. The main goal of this paper is to improve the above discontinuous Galerkin finite element method so that it can handle nonhomogeneous Dirichlet boundary conditions effectively. In addition, the method has been generalized in terms of approximation of the weak gradient. Error estimates of optimal order are established for the corresponding discontinuous finite element approximation in both a discrete H 1 norm and the L 2 norm. Numerical results are presented to confirm the theory.
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The capability to incorporate moving geometric features within models for complex simulations is a common requirement in many fields. The fluid mechanics within aeronautical applications, for example, routinely feature rotating (e.g. turbines, wheels and fan blades) or sliding components (e.g. in compressor or turbine cascade simulations). With an increasing trend towards the high-fidelity modelling of these cases, in particular combined with the use of high-order discontinuous Galerkin methods, there is therefore a requirement to understand how different numerical treatments of the interfaces between the static mesh and the sliding/rotating part impact on overall solution quality. In this article, we compare two different approaches to handle this non-conformal interface. The first is the so-called mortar approach, where flux integrals along edges are split according to the positioning of the non-conformal grid. The second is a lesser-documented point-to-point interpolation method,...
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SIAM Journal on Scientific Computing, 2008
A postprocessing technique based on negative order norm estimates for the discontinuous Galerkin methods was previously introduced by Cockburn, Luskin, Shu, and Süli []. The postprocessor allows improvement in accuracy of the discontinuous Galerkin method for time-dependent linear hyperbolic equations from order k+1 to order 2k+1 over a uniform mesh. Assumptions on the convolution kernel along with uniformity in mesh size give a local translation invariant postprocessor that allows for simple implementation using small matrix-vector multiplications. In this paper, we present two alternatives for extending this postprocessing technique to include smoothly varying meshes. The first method uses a simple local L 2 -projection of the smoothly varying mesh to a locally uniform mesh and uses this projected solution to compute the postprocessed solution. By using this local L 2 -projection, recalculating the convolution kernel for every element can be avoided, and 2k+1 order accuracy of the postprocessed solution can be achieved. The second method uses the idea of characteristic length based upon the largest element size for the scaling of the postprocessing kernel. These two methods, local projection and characteristic length, are also applied to approximations over a mesh with elements that vary in size randomly. We discuss the computational issues in using these two techniques and demonstrate numerically that we obtain the 2k+1 order of accuracy for the smoothly varying meshes, and that although the 2k+1 order of accuracy is not fully realized for random meshes, there is significant improvement in the L 2 -errors.
2003
This paper is a short essay on discontinuous Galerkin methods intended for a very wide audience. We present the discontinuous Galerkin methods and describe and discuss their main features. Since the methods use completely discontinuous approximations, they produce mass matrices that are block-diagonal. This renders the methods highly parallelizable when applied to hyperbolic problems. Another consequence of the use of discontinuous approximations is that these methods can easily handle irregular meshes with hanging nodes and approximations that have polynomials of different degrees in different elements. They are thus ideal for use with adaptive algorithms. Moreover, the methods are locally conservative (a property highly valued by the computational fluid dynamics community) and, in spite of providing discontinuous approximations, stable, and high-order accurate. Even more, when applied to non-linear hyperbolic problems, the discontinuous Galerkin methods are able to capture highly ...
Mathematics and Computers in Simulation, 2015
We present a high-order discontinuous Galerkin (DG) method for solving the time-dependent Maxwell equations on non-conforming hybrid meshes. The hybrid mesh combines unstructured tetrahedra for the discretization of irregularly shaped objects with a hexahedral mesh for the rest of the computational domain. The transition between tetrahedra and hexahedra is completely non-conform, that is, no pyramidal or prismatic elements are introduced to link these elements. Within each mesh element, the electromagnetic field components are approximated by a arbitrary order nodal polynomial and a centered approximation is used for the evaluation of numerical fluxes at inter-element boundaries. The time integration of the associated semi-discrete equations is achieved by a fourth-order leapfrog scheme. The method is described and discussed, including algorithm formulation, stability, and practical implementation issues such as the hybrid mesh generation and the computation of flux matrices with cubature rules. We illustrate the performance of the proposed method on several two-and three-dimensional examples involving comparisons with DG methods on single element-type meshes. The results show that the use of non-conforming hybrid meshes in DG methods allows for a notable reduction in computing time without sacrificing accuracy.
IEEE Transactions on Magnetics, 2000
This paper is concerned with the design of a high-order discontinuous Galerkin (DG) method for solving the 2-D time-domain Maxwell equations on nonconforming triangular meshes. The proposed DG method allows for using nonconforming meshes with arbitrary-level hanging nodes. This method combines a centered approximation for the evaluation of fluxes at the interface between neighboring elements of the mesh, with a leap-frog time integration scheme. Numerical experiments are presented which both validate the theoretical results and provide further insights regarding to the practical performance of the proposed DG method, particulary when nonconforming meshes are employed.