An official website of the United States government
Surface Stokes and Navier–Stokes equations are used to model fluid flow on surfaces. They have attracted significant recent attention in the numerical analysis literature because approximation of their solutions poses significant challenges not encountered in the Euclidean context. One challenge comes from the need to simultaneously enforce tangentiality and $H^1$ conformity (continuity) of discrete vector fields used to approximate solutions in the velocity-pressure formulation. Existing methods in the literature all enforce one of these two constraints weakly either by penalization or by use of Lagrange multipliers. Missing so far is a robust and systematic construction of surface Stokes finite element spaces which employ nodal degrees of freedom, including MINI, Taylor–Hood, Scott–Vogelius, and other composite elements which can lead to divergence-conforming or pressure-robust discretizations. In this paper we construct surface MINI spaces whose velocity fields are tangential. They are not $H^1$-conforming, but do lie in $H(div)$ and do not require penalization to achieve optimal convergence rates. We prove stability and optimal-order energy-norm convergence of the method and demonstrate optimal-order convergence of the velocity field in $$L_2$$ via numerical experiments. The core advance in the paper is the construction of nodal degrees of freedom for the velocity field. This technique also may be used to construct surface counterparts to many other standard Euclidean Stokes spaces, and we accordingly present numerical experiments indicating optimal-order convergence of nonconforming tangential surface Taylor–Hood elements.
more »
« less
Higher order divergence-free and curl-free interpolation on MAC grids
Divergence-free vector fields and curl-free vector fields play an important role in many types of problems, including the incompressible Navier-Stokes equations, Maxwell's equations, the equations for magnetohydrodynamics, and surface reconstruction. In practice, these fields are often obtained by projection, resulting in a discrete approximation of the continuous field that is discretely divergence-free or discretely curl-free. This field can then be interpolated to non-grid locations, which is required for many algorithms such as particle tracing or semi-Lagrangian advection. This interpolated field will not generally be divergence-free or curl-free in the analytic sense. In this work, we assume these fields are stored on a MAC grid layout and that the divergence and curl operators are discretized using finite differences. This work builds on and extends [39] in multiple ways: (1) we design a divergence-free interpolation scheme that preserves the discrete flux, (2) we adapt the general construction of divergence-free fields into a general construction for curl-free fields, (3) we extend the framework to a more general class of finite difference discretizations, and (4) we use this flexibility to construct fourth-order accurate interpolation schemes for the divergence-free case and the curl-free case. All of the constructions and specific schemes are explicit piecewise polynomials over a local neighborhood.
more »
« less
- Award ID(s):
- 2006570
- PAR ID:
- 10558594
- Publisher / Repository:
- Elsevier
- Date Published:
- Journal Name:
- Journal of Computational Physics
- Volume:
- 503
- Issue:
- C
- ISSN:
- 0021-9991
- Page Range / eLocation ID:
- 112831
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
This work is part of a broader project to investigate student understanding of mathematical ideas used in upper-division physics. This study in particular probes students’ understanding of the divergence and curl operators as applied to vector field diagrams. We examined how students reason with partial derivatives that constitute divergence and curl of the vector field diagrams. Students’ written responses to a task on derivatives, divergence, and curl of a 2D vector field were collected and coded. Students were generally successful in determining the sign of some of the constituent derivatives of div and curl, but struggled in one case in which components were negative. Analysis of written explanations showed confusion between the sign, direction, and change in the magnitude of vector field components.more » « less
-
Abstract Immersed boundary methods are high-order accurate computational tools used to model geometrically complex problems in computational mechanics. While traditional finite element methods require the construction of high-quality boundary-fitted meshes, immersed boundary methods instead embed the computational domain in a structured background grid. Interpolation-based immersed boundary methods augment existing finite element software to non-invasively implement immersed boundary capabilities through extraction. Extraction interpolates the structured background basis as a linear combination of Lagrange polynomials defined on a foreground mesh, creating an interpolated basis that can be easily integrated by existing methods. This work extends the interpolation-based immersed isogeometric method to multi-material and multi-physics problems. Beginning from level-set descriptions of domain geometries, Heaviside enrichment is implemented to accommodate discontinuities in state variable fields across material interfaces. Adaptive refinement with truncated hierarchically refined B-splines (THB-splines) is used to both improve interface geometry representations and to resolve large solution gradients near interfaces. Multi-physics problems typically involve coupled fields where each field has unique discretization requirements. This work presents a novel discretization method for coupled problems through the application of extraction, using a single foreground mesh for all fields. Numerical examples illustrate optimal convergence rates for this method in both 2D and 3D, for partial differential equations representing heat conduction, linear elasticity, and a coupled thermo-mechanical problem. The utility of this method is demonstrated through image-based analysis of a composite sample, where in addition to circumventing typical meshing difficulties, this method reduces the required degrees of freedom when compared to classical boundary-fitted finite element methods.more » « less
-
Dreyfus, T; González-Martín, A S; Nardi, E; Monaghan, J; Thompson, P W (Ed.)Physics students are introduced to vector fields in introductory courses, typically in the contexts of electric and magnetic fields. Vector calculus provides several ways to describe how vector fields vary in space including the gradient, divergence, and curl. Physics majors use vector calculus extensively in a junior-level electricity and magnetism (E&M) course. Our focus here is exploring student reasoning with the partial derivatives that constitute divergence and curl in vector field representations, adding to the current understanding of how students reason with derivatives.more » « less
-
Abstract Adaptive mesh refinement (AMR) is the art of solving PDEs on a mesh hierarchy with increasing mesh refinement at each level of the hierarchy. Accurate treatment on AMR hierarchies requires accurate prolongation of the solution from a coarse mesh to a newly defined finer mesh. For scalar variables, suitably high-order finite volume WENO methods can carry out such a prolongation. However, classes of PDEs, such as computational electrodynamics (CED) and magnetohydrodynamics (MHD), require that vector fields preserve a divergence constraint. The primal variables in such schemes consist of normal components of the vector field that are collocated at the faces of the mesh. As a result, the reconstruction and prolongation strategies for divergence constraint-preserving vector fields are necessarily more intricate. In this paper we present a fourth-order divergence constraint-preserving prolongation strategy that is analytically exact. Extension to higher orders using analytically exact methods is very challenging. To overcome that challenge, a novel WENO-like reconstruction strategy is invented that matches the moments of the vector field in the faces, where the vector field components are collocated. This approach is almost divergence constraint-preserving, therefore, we call it WENO-ADP. To make it exactly divergence constraint-preserving, a touch-up procedure is developed that is based on a constrained least squares (CLSQ) method for restoring the divergence constraint up to machine accuracy. With the touch-up, it is called WENO-ADPT. It is shown that refinement ratios of two and higher can be accommodated. An item of broader interest in this work is that we have also been able to invent very efficient finite volume WENO methods, where the coefficients are very easily obtained and the multidimensional smoothness indicators can be expressed as perfect squares. We demonstrate that the divergence constraint-preserving strategy works at several high orders for divergence-free vector fields as well as vector fields, where the divergence of the vector field has to match a charge density and its higher moments. We also show that our methods overcome the late time instability that has been known to plague adaptive computations in CED.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1016/j.jcp.2024.112831
