Artur Palha

Physics-compatible discretization techniques on single and dual grids, with application to the Poisson equation of volume forms

This paper introduces the basic concepts for physics-compatible discretization techniques. The paper gives a clear distinction between vectors and forms. Based on the difference between forms and pseudo-forms and the @?-operator which switches between the two, a dual grid description and a single grid description are presented. The dual grid method resembles a staggered finite volume method, whereas the single grid approach shows a strong resemblance with a finite element method. Both approaches are compared for the Poisson equation for volume forms. By defining a suitably weighted inner product for 1-forms this approach can readily be applied to anisotropic diffusion models for volume forms.

Mimetic least-squares spectral/ hp finite element method for the poisson equation

Mimetic approaches to the solution of partial differential equations (PDEs) produce numerical schemes which are compatible with the structural properties – conservation of certain quantities and symmetries, for example – of the systems being modelled Least Squares (LS) schemes offer many desirable properties, most notably the fact that they lead to symmetric positive definite algebraic systems, which represent an advantage in terms of computational efficiency of the scheme Nevertheless, LS methods are known to lack proper conservation properties which means that a mimetic formulation of LS, which guarantees the conservation properties, is of great importance In the present work, the LS approach appears in order to minimise the error between the dual variables, implementing weakly the material laws, obtaining an optimal approximation for both variables The application to a 2D Poisson problem and a comparison will be made with a standard LS finite element scheme.

A Conservative Spectral Element Method for Curvilinear Domains

This paper describes a mimetic spectral element method on curvilinear grids applied to the Poisson equation. The Poisson equation is formulated in terms of differential forms. The spectral basis functions in which the differential forms are expressed lead to a metric free discrete representation of the gradient and the divergence operator. Using the fact that the pullback operator commutes with the wedge product and the exterior derivative leads to a mimetic spectral element formulation on curvilinear grids which displays exponential convergence and satisfies the divergence exactly. The robustness of the proposed scheme will be demonstrated for a sample problem for which exponential convergence is obtained.

Spectral Element Approximation of the Hodge-⋆ Operator in Curved Elements

Mimetic approaches to the solution of partial differential equations (PDE’s) produce numerical schemes which are compatible with the structural properties – conservation of certain quantities and symmetries, for example – of the systems being modelled. Least Squares (LS) schemes offer many desirable properties, most notably the fact that they lead to symmetric positive definite algebraic systems, which represent an advantage in terms of computational efficiency of the scheme. Nevertheless, LS methods are known to lack proper conservation properties which means that a mimetic formulation of LS, which guarantees the conservation properties, is of great importance. In the present work, the LS approach appears in order to minimize the error between the dual variables, implementing weakly the material laws, obtaining an optimal approximation for both variables. The application to a 2D Poisson problem and a comparison will be made with a standard LS finite element scheme, see, for example, Cai et al. (J. Numer. Anal. 34:425–454, 1997).

Least-Squares spectral element method on a staggered grid

This paper describes a mimetic spectral element formulation for the Poisson equation on quadrilateral elements Two dual grids are employed to represent the two first order equations The discrete Hodge operator, which connects variables on these two grids, is the derived Hodge operator obtained from the wedge product and the inner-product The gradient operator is not discretized directly, but derived from the symmetry relation between gradient and divergence on dual meshes, as proposed by Hyman et al., [5], thus ensuring a symmetric discrete Laplace operator The resulting scheme is a staggered spectral element scheme, similar to the staggering proposed by Kopriva and Kolias, [6] Different integration schemes are used for the various terms involved This scheme is equivalent to a least-squares formulation which minimizes the difference between the dual velocity representations This generates the discrete Hodge-⋆ operator The discretization error of this schemes equals the interpolation error.

The Geometric Basis of Numerical Methods

The relation between physics, its description in terms of partial differential equations and geometry is explored in this paper. Geometry determines the correct weak formulation in finite element methods and also dictates which basis functions should be employed to obtain discrete well-posedness.

Mixed Mimetic Spectral Element Method Applied to Darcy’s Problem

We present a discretization for Darcy’s problem using the recently developed Mimetic Spectral Element Method (Kreeft et al. (2011) Mimetic framework on curvilinear quadrilaterals of arbitrary order. Submitted to FoCM, Arxiv preprint arXiv:1111.4304). The gist lies in the exact discrete representation of integral relations. In this paper, an anisotropic flow through a porous medium is considered and a discretization of a full permeability tensor is presented. The performance of the method is evaluated on standard test problems, converging at the same rate as the best possible approximation.

Mimetic Spectral Element Advection

We present a discretization of the linear advection of differential forms on bounded domains. The framework established in [4] is extended to incorporate the Lie derivative, \(\mathcal{L}\), by means of Cartan’s homotopy formula. The method is based on a physics-compatible discretization with spectral accuracy. It will be shown that the derived scheme has spectral convergence with local mass conservation. Artificial dispersion depends on the order of time integration.

Spectral Mimetic Least-Squares Method for Curl-curl Systems

One of the most cited disadvantages of least-squares formulations is its lack of conservation. By a suitable choice of least-squares functional and the use of appropriate conforming finite dimensional function spaces, this drawback can be completely removed. Such a mimetic least-squares method is applied to a curl-curl system. Conservation properties will be proved and demonstrated by test results on two-dimensional curvilinear grids.