Mimetic Spectral Element Method for Anisotropic Diffusion

Lecture Notes in Computational Science and Engineering

et al. 2020

Self Cite

Hybrid formulations [1, 3, 10] are classical domain decomposition methods which reduce the problem of solving one global system to many small local systems. The local systems can then be efficiently solved independently of each other in parallel. In this work we present a hybrid mimetic spectral element formulation to solve Darcy flow. We follow [8] which render the constraints on divergence of mass flux, the pressure gradient and the inter-element continuity metric free. The resulting system is extremely sparse and shows a reduced growth in condition number as compared to a non-hybrid system. This document is structured as follows: In Sect. 2 we define the weak formulation for Darcy flow. The basis functions are introduced in Sect. 3. The evaluation of weighted inner product and duality pairings are discussed in Sect. 4. In Sect. 5 we discuss the formulation of discrete algebraic system. In Sect. 6 we present results for a test case taken from [7].

Section: Resultsmentioning

confidence: 90%

Section: Resultsmentioning

confidence: 99%

Section: Divergence Of Velocitymentioning

confidence: 99%

“…E N is the assembled N for all elements. For, K = 2 × 2, N = 2, E N is shown in (6). The matrix size of E N is 8 × 64, but it has only 16 non-zero entities.…”

Section: Connectivity Matrixmentioning

confidence: 99%

See 2 more Smart Citations

A Conservative Hybrid Method for Darcy Flow

Jain

Fisser

Lecture Notes in Computational Science and Engineering

et al. 2020

Self Cite

“…This work extends [7,9], where similar dual Neumann-Dirichlet problems are considered, to 3-dimensional space. These primal spaces and their algebraic dual representations can be ideal for the so-called mimetic or structure-preserving discretizations [1,4,8,11,12]. Together with their trace spaces, they can be used for the hybrid finite element methods which first decompose the domains into discontinuous elements then connect them with Lagrange multipliers living in the trace spaces [2,13,14].…”

Section: Introductionmentioning

confidence: 99%

Discrete Equivalence of Adjoint Neumann–Dirichlet div-grad and grad-div Equations in Curvilinear 3D Domains

Zhang

Jain

Lecture Notes in Computational Science and Engineering

et al. 2020

Self Cite

In this paper, we will show that the equivalence of a div-grad Neumann problem and a grad-div Dirichlet problem can be preserved at the discrete level in 3-dimensional curvilinear domains if algebraic dual polynomial representations are employed. These representations will be introduced. Proof of the equivalence at the discrete level follows from the construction of the algebraic dual representations. A 3-dimensional test problem in curvilinear coordinates will illustrate this approach.

A mixed mimetic spectral element model of the 3D compressible Euler equations on the cubed sphere

Lee

Journal of Computational Physics

2020

A model of the three-dimensional rotating compressible Euler equations on the cubed sphere is presented. The model uses a mixed mimetic spectral element discretization which allows for the exact exchanges of kinetic, internal and potential energy via the compatibility properties of the chosen function spaces. A Strang carryover dimensional splitting procedure is used, with the horizontal dynamics solved explicitly and the vertical dynamics solved implicitly so as to avoid the CFL restriction of the vertical sound waves. The function spaces used to represent the horizontal dynamics are discontinuous across vertical element boundaries, such that each horizontal layer is solved independently so as to avoid the need to invert a global 3D mass matrix, while the function spaces used to represent the vertical dynamics are similarly discontinuous across horizontal element boundaries, allowing for the serial solution of the vertical dynamics independently for each horizontal element. The model is validated against standard test cases for baroclinic instability within an otherwise hydrostatically and geostrophically balanced atmosphere, and a nonhydrostatic gravity wave as driven by a temperature perturbation.