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Volume 32, Issue 1
Continuous Finite Element Subgrid Basis Functions for Discontinuous Galerkin Schemes on Unstructured Polygonal Voronoi Meshes

Walter Boscheri, Michael Dumbser & Elena Gaburro

Commun. Comput. Phys., 32 (2022), pp. 259-298.

Published online: 2022-07

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  • Abstract

We propose a new high order accurate nodal discontinuous Galerkin (DG) method for the solution of nonlinear hyperbolic systems of partial differential equations (PDE) on unstructured polygonal Voronoi meshes. Rather than using classical polynomials of degree $N$ inside each element, in our new approach the discrete solution is represented by piecewise continuous polynomials of degree $N$ within each Voronoi element, using a continuous finite element basis defined on a subgrid inside each polygon. We call the resulting subgrid basis an agglomerated finite element (AFE) basis for the DG method on general polygons, since it is obtained by the agglomeration of the finite element basis functions associated with the subgrid triangles. The basis functions on each sub-triangle are defined, as usual, on a universal reference element, hence allowing to compute universal mass, flux and stiffness matrices for the subgrid triangles once and for all in a pre-processing stage for the reference element only. Consequently, the construction of an efficient quadrature-free algorithm is possible, despite the unstructured nature of the computational grid. High order of accuracy in time is achieved thanks to the ADER approach, making use of an element-local space-time Galerkin finite element predictor.
The novel schemes are carefully validated against a set of typical benchmark problems for the compressible Euler and Navier-Stokes equations. The numerical results have been checked with reference solutions available in literature and also systematically compared, in terms of computational efficiency and accuracy, with those obtained by the corresponding modal DG version of the scheme.

  • Keywords

Continuous finite element subgrid basis for DG schemes, high order quadrature-free ADER-DG schemes, unstructured Voronoi meshes, comparison of nodal and modal basis, compressible Euler and Navier-Stokes equations.

  • AMS Subject Headings

65Mxx, 65Yxx

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-32-259, author = {Walter and Boscheri and and 24063 and and Walter Boscheri and Michael and Dumbser and and 24064 and and Michael Dumbser and Elena and Gaburro and and 24065 and and Elena Gaburro}, title = {Continuous Finite Element Subgrid Basis Functions for Discontinuous Galerkin Schemes on Unstructured Polygonal Voronoi Meshes}, journal = {Communications in Computational Physics}, year = {2022}, volume = {32}, number = {1}, pages = {259--298}, abstract = {

We propose a new high order accurate nodal discontinuous Galerkin (DG) method for the solution of nonlinear hyperbolic systems of partial differential equations (PDE) on unstructured polygonal Voronoi meshes. Rather than using classical polynomials of degree $N$ inside each element, in our new approach the discrete solution is represented by piecewise continuous polynomials of degree $N$ within each Voronoi element, using a continuous finite element basis defined on a subgrid inside each polygon. We call the resulting subgrid basis an agglomerated finite element (AFE) basis for the DG method on general polygons, since it is obtained by the agglomeration of the finite element basis functions associated with the subgrid triangles. The basis functions on each sub-triangle are defined, as usual, on a universal reference element, hence allowing to compute universal mass, flux and stiffness matrices for the subgrid triangles once and for all in a pre-processing stage for the reference element only. Consequently, the construction of an efficient quadrature-free algorithm is possible, despite the unstructured nature of the computational grid. High order of accuracy in time is achieved thanks to the ADER approach, making use of an element-local space-time Galerkin finite element predictor.
The novel schemes are carefully validated against a set of typical benchmark problems for the compressible Euler and Navier-Stokes equations. The numerical results have been checked with reference solutions available in literature and also systematically compared, in terms of computational efficiency and accuracy, with those obtained by the corresponding modal DG version of the scheme.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2021-0235}, url = {http://global-sci.org/intro/article_detail/cicp/20794.html} }
TY - JOUR T1 - Continuous Finite Element Subgrid Basis Functions for Discontinuous Galerkin Schemes on Unstructured Polygonal Voronoi Meshes AU - Boscheri , Walter AU - Dumbser , Michael AU - Gaburro , Elena JO - Communications in Computational Physics VL - 1 SP - 259 EP - 298 PY - 2022 DA - 2022/07 SN - 32 DO - http://doi.org/10.4208/cicp.OA-2021-0235 UR - https://global-sci.org/intro/article_detail/cicp/20794.html KW - Continuous finite element subgrid basis for DG schemes, high order quadrature-free ADER-DG schemes, unstructured Voronoi meshes, comparison of nodal and modal basis, compressible Euler and Navier-Stokes equations. AB -

We propose a new high order accurate nodal discontinuous Galerkin (DG) method for the solution of nonlinear hyperbolic systems of partial differential equations (PDE) on unstructured polygonal Voronoi meshes. Rather than using classical polynomials of degree $N$ inside each element, in our new approach the discrete solution is represented by piecewise continuous polynomials of degree $N$ within each Voronoi element, using a continuous finite element basis defined on a subgrid inside each polygon. We call the resulting subgrid basis an agglomerated finite element (AFE) basis for the DG method on general polygons, since it is obtained by the agglomeration of the finite element basis functions associated with the subgrid triangles. The basis functions on each sub-triangle are defined, as usual, on a universal reference element, hence allowing to compute universal mass, flux and stiffness matrices for the subgrid triangles once and for all in a pre-processing stage for the reference element only. Consequently, the construction of an efficient quadrature-free algorithm is possible, despite the unstructured nature of the computational grid. High order of accuracy in time is achieved thanks to the ADER approach, making use of an element-local space-time Galerkin finite element predictor.
The novel schemes are carefully validated against a set of typical benchmark problems for the compressible Euler and Navier-Stokes equations. The numerical results have been checked with reference solutions available in literature and also systematically compared, in terms of computational efficiency and accuracy, with those obtained by the corresponding modal DG version of the scheme.

Walter Boscheri, Michael Dumbser & Elena Gaburro. (2022). Continuous Finite Element Subgrid Basis Functions for Discontinuous Galerkin Schemes on Unstructured Polygonal Voronoi Meshes. Communications in Computational Physics. 32 (1). 259-298. doi:10.4208/cicp.OA-2021-0235
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