Smoothness-Increasing Accuracy-Conserving Filters for Discontinuous Galerkin Solutions over Unstructured Triangular Meshes

Abstract

Abstract. The discontinuous Galerkin (DG) method has very quickly found utility in such diverse applications as computational solid mechanics, fluid mechanics, acoustics, and electromagnetics. The DG methodology merely requires weak constraints on the fluxes between elements. This feature provides a flexibility which is difficult to match with conventional continuous Galerkin methods. However, allowing discontinuity between element interfaces can in turn be problematic during simulation postprocessing, such as in visualization. Consequently, smoothness-increasing accuracyconserving (SIAC) filters were proposed in [M. Steffen et al., IEEE Trans. Vis. Comput. Graph., 14 (2008), pp. 680-692, D. Walfisch et al., J. Sci. Comput., 38 (2009 as a means of introducing continuity at element interfaces while maintaining the order of accuracy of the original input DG solution. Although the DG methodology can be applied to arbitrary triangulations, the typical application of SIAC filters has been to DG solutions obtained over translation invariant meshes such as structured quadrilaterals and triangles. As the assumption of any sort of regularity including the translation invariance of the mesh is a hindrance towards making the SIAC filter applicable to real life simulations, we demonstrate in this paper for the first time the behavior and complexity of the computational extension of this filtering technique to fully unstructured tessellations. We consider different types of unstructured triangulations and show that it is indeed possible to get reduced errors and improved smoothness through a proper choice of kernel scaling. These results are promising, as they pave the way towards a more generalized SIAC filtering technique.

Exploiting Superconvergence Through Smoothness-Increasing Accuracy-Conserving (SIAC) Filtering

“…This is illustrated in Figure 5. In this figure, a kernel scaling of H = mh is used for the convolution kernel in the SIAC filter for a discontinuous Galerkin approximation over a uniform mesh designated by h. We can see that error reduction actually occurs even when H < h. Superconvergent order starts to occur around H = h and errors start to increase for H > h. The sweetspot of reduced errors and superconvergence seems to occur around H = h. Although the typical meshes tested involve some type of translation invariance, the SIAC filter has also been tested over unstructured triangular meshes with promising results [10,12]. For example Figure 6 shows the difference in the pointwise errors for the DG approximation versus the SIAC filtered DG approximation.…”

On the Treatment of Field Quantities and Elemental Continuity in FEM Solutions

“…Although the characteristic length choice is obvious for uniform meshes (H = mesh size), there has been ongoing work on finding the optimal value for non-uniform and unstructured meshes [4,17,26]. In addition, introducing a rotation in the kernel leads to the problem of determining the optimal orientation.…”

“…Regarding the kernel characteristic length, [14,17] used the formula H = m · h, with h being the largest element size, and performed global error analyses on known analytic fields for different values of m. They showed that asymptotically, the value m = 1 lead to optimal results. We performed a similar experiment and in Figure 11 we show the same filtered vortex after applying different characteristic lengths where one can appreciate that the value H = 0.675 (m = 1) gives the most satisfactory results.…”

“…• The minimum error occurs when LSIAC's characteristic length is equal to maximum edge length (i.e., m = 1.0) as stated in [14,17].…”

Smoothness-Increasing Accuracy-Conserving (SIAC) Filtering and Quasi-Interpolation: A Unified View

“…A SIAC filter has the ability to extract a superconvergent solution from a DG approximation for different element types including quadrilateral, structured triangular, tetrahedral and even unstructured triangular meshes [17,21,18]. One-sided SIAC kernels have been proposed as an extension of this convolution-based postprocessing for simulations involving boundaries or sharp discontinuities such as shocks [24,35,26].…”