3rd and 11th orders of accuracy of ‘linear’ and ‘quadratic’ elements for Poisson equation with irregular interfaces on Cartesian meshes

Purpose The purpose of this paper is as follows: to significantly reduce the computation time (by a factor of 1,000 and more) compared to known numerical techniques for real-world problems with complex interfaces; and to simplify the solution by using trivial unfitted Cartesian meshes (no need in complicated mesh generators for complex geometry). Design/methodology/approach This study extends the recently developed optimal local truncation error method (OLTEM) for the Poisson equation with constant coefficients to a much more general case of discontinuous coefficients that can be applied to domains with different material properties (e.g. different inclusions, multi-material structural components, etc.). This study develops OLTEM using compact 9-point and 25-point stencils that are similar to those for linear and quadratic finite elements. In contrast to finite elements and other known numerical techniques for interface problems with conformed and unfitted meshes, OLTEM with 9-point and 25-point stencils and unfitted Cartesian meshes provides the 3-rd and 11-th order of accuracy for irregular interfaces, respectively; i.e. a huge increase in accuracy by eight orders for the new 'quadratic' elements compared to known techniques at similar computational costs. There are no unknowns on interfaces between different materials; the structure of the global discrete system is the same for homogeneous and heterogeneous materials (the difference in the values of the stencil coefficients). The calculation of the unknown stencil coefficients is based on the minimization of the local truncation error of the stencil equations and yields the optimal order of accuracy of OLTEM at a given stencil width. The numerical results with irregular interfaces show that at the same number of degrees of freedom, OLTEM with the 9-points stencils is even more accurate than the 4-th order finite elements; OLTEM with the 25-points stencils is much more accurate than the 7-th order finite elements with much wider stencils and conformed meshes. Findings The significant increase in accuracy for OLTEM by one order for 'linear' elements and by 8 orders for 'quadratic' elements compared to that for known techniques. This will lead to a huge reduction in the computation time for the problems with complex irregular interfaces. The use of trivial unfitted Cartesian meshes significantly simplifies the solution and reduces the time for the data preparation (no need in complicated mesh generators for complex geometry). Originality/value It has been never seen in the literature such a huge increase in accuracy for the proposed technique compared to existing methods. Due to a high accuracy, the proposed technique will allow the direct solution of multiscale problems without the scale separation.

$$H^1$$-conforming finite element cochain complexes and commuting quasi-interpolation operators on Cartesian meshes

A finite element cochain complex on Cartesian meshes of any dimension based on the $$H^1$$ H 1 -inner product is introduced. It yields $$H^1$$ H 1 -conforming finite element spaces with exterior derivatives in $$H^1$$ H 1 . We use a tensor product construction to obtain $$L^2$$ L 2 -stable projectors into these spaces which commute with the exterior derivative. The finite element complex is generalized to a family of arbitrary order.

A unified first order hyperbolic model for nonlinear dynamic rupture processes in diffuse fracture zones

<p>Earthquake fault zones are more complex, both geometrically and rheologically, than an idealised infinitely thin plane embedded in linear elastic material.  Field and laboratory measurements reveal complex fault zone structure involving tensile and shear fractures spanning a wide spectrum of length scales (e.g., Mitchell & Faulkner, 2009), dense seismic and geodetic recording of small and large earthquakes show hierarchical volumetric faulting patterns (e.g., Cheng et al., 2018, Ross et al., 2019) and 2D numerical models explicitly accounting for off-fault fractures demonstrate important feedback with rupture dynamics and ground motions (e.g., Thomas & Bhat 2018, Okubo et al., 2019).</p><p>Here (Gabriel et al., 2021) we adopt a diffuse crack representation to incorporate finite strain nonlinear material behaviour, natural complexities and multi-physics coupling within and outside of fault zones into dynamic earthquake rupture modeling. We use a first-order hyperbolic and thermodynamically compatible mathematical model, namely the GPR model (Godunov & Romenski, 1972; Romenski, 1988),  to describe a continuum in a gravitational field which provides a unified description of nonlinear elasto-plasticity, material damage and of viscous Newtonian flows with phase transition between solid and liquid phases.</p><p>The model shares common features with phase-field approaches but substantially extends them. Pre-damaged faults as well as dynamically induced secondary cracks are therein described via a scalar function indicating the local level of material damage (Tavelli et al., 2020); arbitrarily complex geometries are represented via a diffuse interface approach based on a solid volume fraction function (Tavelli et al., 2019). Neither of the two scalar fields needs to be mesh-aligned, allowing thus faults and cracks with complex topology and the use of adaptive Cartesian meshes (AMR). High-order accuracy and adaptive Cartesian meshes are enabled in 2D and 3D by using the extreme scale hyperbolic PDE solver ExaHyPE (Reinarz et al., 2019).</p><p>We show a wide range of numerical applications that are relevant for dynamic earthquake rupture in fault zones, including the co-seismic generation of secondary off-fault shear cracks, tensile rock fracture in the Brazilian disc test, as well as a natural convection problem in molten rock-like material. We compare diffuse interface fault models of kinematic cracks, spontaneous dynamic rupture and dynamically generated off-fault shear cracks to sharp interface reference models. To this end, we calibrate the GPR model to resemble empirical tensile and shear crack formation and friction laws. We find that the continuum model can resemble and extend classical solutions, while introducing dynamic differences (i) on the scale of pre-damaged/low-rigidity fault zone, such as out-of- plane rupture rotation; and (ii) on the scale of the intact host rock, such as conjugate shear cracking in tensile lobes. </p><p>Our approach is part of the TEAR ERC project (www.tear-erc.eu) and will potentially allow to fully model volumetric fault zone shearing during earthquake rupture, which includes spontaneous partition of fault slip into intensely localized shear deformation within weaker (possibly cohesionless/ultracataclastic) fault-core gouge and more distributed damage within fault rocks and foliated gouges.</p>

2D tracer transport on Cartesian and icosahedral grids: scheme comparisons for idealized test cases

<p>The distribution of tracers in the atmosphere results from the presence and emission of gaseous and particulate matter, as well as their transport, sedimentation and (photo-)chemical transformations. Understanding and quantifying these processes in the atmosphere can be addressed through the use of global-scale or regional-scale chemistry-transport numerical models such as CHIMERE (Mailler et al., 2016).</p><p> While possible in principle, it is impractical to use this model to represent long-range transport of dense plumes of gas and aerosols, resulting for instance from massive emissions by volcanic eruptions, forest fires and desertic aerosol tempests. Indeed such studies requiring both large domains and high resolution have a prohibitive numerical cost due to the formulation of CHIMERE on a regular Cartesian mesh. This limitation is shared by all currently operational chemistry-transport models. Additionally, traditional Cartesian meshes pose a numerical singularity at the poles, where the longitude lines converge.</p><p> These limitations may be lifted by replacing CHIMERE’s Cartesian mesh by a fully unstructured mesh. This would allow modelers to vary resolution in space, and hence to focus computational resources in key regions with sharp variations (e.g. volcanic eruptions) where high spatial and temporal resolution is required.</p><p> As a first step in this direction, we compare the numerical performance of transport schemes formulated on Cartesian meshes and schemes formulated on unstructured meshes (Dubey et al., 2015). To focus on differences due to numerics, the unstructured mesh is a quasi-uniform icosahedral mesh such as the one used by global dynamical core DYNAMICO (Dubos et al., 2015). Spatial and temporal coupled and de-coupled schemes of various order are implemented in each mesh framework. A suite of test cases is used to evaluate different properties of the mesh-scheme pairings.To avoid the Cartesian pole singularity, the Cartesian mesh covers a limited domain excluding the poles. Analytical wind fields adapted to this limited domain are used. Metrics are evaluated using the quantities obtained in the simulations, such as convergence using root mean square errors, shape preservation using non-linear tracer relations, and diffusion using total entropy. The stability and monotonicity of the used schemes are also numerically validated.</p><p> We find that a scheme of the Van Leer family on the unstructured mesh has a performance slightly inferior to a similar scheme on a Cartesian mesh. However, since this loss in quality remains moderate, it should be possible to more than compensate for it with a variable resolution. We are currently investigating this question and will present variable-resolution results if this ongoing work is timely completed. If successful, fully unstructured meshes would be a significant step forward in the modeling of scale interactions in atmospheric chemistry, and would potentially allow breakthrough for the understanding of such interactions.</p>