08:30
Error Estimation and Analysis (Part 3)
Chair: Fred Wubs
08:30
25 mins
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On mesh regularity conditions for simplicial finite elements
Sergey Korotov, Jon Eivind Vatne, Ali Khademi
Abstract: In this talk, we will present natural and practical generalizations of the minimum and maximum angle
conditions, commonly used in the finite element analysis for triangular and tetrahedral
meshes, to the case of simplicial meshes in any space dimension. The relation (mostly
equivalence) of these conditions to some other mesh regularity conditions and some applications
will be discussed.
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08:55
25 mins
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Doubling the convergence rate by pre- and post-processing the finite element approximation for linear wave problems
Sjoerd Geevers Geevers
Abstract: A novel pre- and post-processing algorithm is presented that can double the convergence rate of finite element approximations for linear wave problems. In particular, it can be shown that a q-step pre- and post-processing algorithm can improve the convergence rate of the finite element approximation from order p+1 to order p+1+q in the L^2-norm and from order p to order p+q in the energy norm, in both cases up to a maximum of order 2p, with p the polynomial degree of the finite element space. The q-step pre- and post-processing algorithms only need to be applied once and require solving at most q linear systems of equations.
The biggest advantage of the proposed method compared to other post-processing methods is that it does not suffer from convergence rate loss when using unstructured meshes. Other advantages are that this new pre- and post-processing method is straightforward to implement, incorporates boundary conditions naturally, and does not lose accuracy near boundaries or strong inhomogeneities in the domain. Numerical examples illustrate the improved accuracy and higher convergence rates when using this method. In particular, they confirm that 2p-order convergence rates in the energy norm are obtained, even when using unstructured meshes or when solving problems involving heterogeneous domains and curved boundaries.
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09:20
25 mins
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The 8T-LE Partition applied to the Barycentric Division of a 3-D Cube
Miguel Angel Padron, Angel Plaza
Abstract: A 3-D cube can be triangulated using the barycentric partition. For this partition a new node has to be inserted in the center of the cube. Joining this point with the four vertices of the faces of the cube six square pyramids are generated. Then these pyramids are subdivided into 2 subtetrahedra by the diagonal of the cube faces. With this procedure the 3 dimensional cube is partitioned into 12 subtetrahedra, all of them similar to each other and acute type. These tetrahedra are Sommerville T3.
Starting from this point, each one these tetrahedra can be subdivided in two subtetrahedra by the second diagonal of the cube faces. In this way, 24 subtetrahedra are generated. All these tetrahedra are trirectangular tetrahedra or cube corner tetrahedra and all of them are similar between them. This kind of tetrahedron is acute type and also is an ortho-simplex. In this particular case, these trirectangular tetrahedra are isosceles since two of the three mutually orthogonal edges passing through the same vertex, are the same length.
Finally, these tetrahedra can also be subdivided into 2 subtetrahedra by three parallel planes to the faces of the cube and intersecting in the barycenter. The number of subtetrahedra generated in this way is 48, and they are path-tetrahedra or right-type tetrahedra. All of them are similar to each another and are also ortho-simplices. This kind of tetrahedra are also isosceles according to the definition given above.
In every step of subdivision of the 3-D cube, the triangulation generated is acute type. This feature is of very importance in the finite element analysis of boundary value problems, and guarantee the validity of the discrete maximum principle when solving the Poisson and some other elliptic equations with various boundary conditions.
We are interested in knowing if the 8T-LE partition applied to the tetrahedra mentioned before, produce tetrahedra in which not only the number of similarity classes is bounded, but also if the tetrahedra generated are acute type. This partition was developed in the latest nineties by Angel Plaza and Graham F. Carey, in which every single tetrahedron is subdivided into 8, in accordance with the division of the skeleton of the triangular faces of the tetrahedron by the 4T-LE partition. For this partition, a previous classification of the tetrahedra is required.
The study of the 8T-LE partition applied to these tetrahedra is of interest because the conversion from an octree-based hexahedral mesh to a tetrahedral mesh is satrightforward.
Keywords: Barycentric partition, 8T-LE partition, similarity classes, acute triangulation
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09:45
25 mins
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Stability problems caused by non-conforming curvilinear interfaces
Oskar Ålund, Jan Nordström
Abstract: We highlight an issue regarding the stability of a family of discretizations on summation-by-parts form, using adjoint interface operators to communicate across non-conforming meshes (see attached file for details).
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