HDG Kernels

HDG kernels and their boundary condition counterparts, HDG Boundary Conditions, are advanced systems that should only be developed by users with a fair amount of finite element experience. For background on hybridization, we encourage the user to read (Cockburn et al., 2009) which presents a unified framework for considering hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for elliptic problems. (Cockburn et al., 2008) presents a single-face hybridizable discontinuous Galerkin (HDG) method for an elliptic problem, in which a non-zero stabilization term is added to only one face of a given element. (Nguyen et al., 2010) presents an HDG method for Stokes flow. (Nguyen et al., 2011) extends HDG to Navier-Stokes. More HDG literature may be found by looking at the research of Bernardo Cockburn, his former postdoc Sander Rhebergen, and Rhebergen's former postdoc Tamas Horvath. Work by Tan Bui-Thanh on upwind HDG methods, like in (Bui-Thanh, 2015) is also worth noting.

A hybridized finite element formulation starts with some primal finite element discretization. Then some continuity property of the finite element space is broken. For instance Raviart-Thomas finite elements may be used to solve a mixed formulation description of a Poisson problem. The Raviart-Thomas elements ensure continuity of the normal component of the vector field across element faces. We break that continuity in the finite element space used in the hybridized method and instead introduce degrees of freedom, that live only on the mesh skeleton (the faces of the mesh), that are responsible for ensuring the continuity that was lost by breaking the finite element space. In libMesh/MOOSE implementation terms, when hybridizing the Raviart-Thomas description of the Poisson problem, we change from using a RAVIART_THOMAS basis to an L2_RAVIART_THOMAS basis and introduce a SIDE_HIERARCHIC variable whose degrees of freedom live on the mesh skeleton. We will refer to the variables that exist "before" the hybridization as primal variables and the variable(s) that live on the mesh skeleton as Lagrange multipliers (LMs) or dual variable(s).

We note that some classes of HDG methods, such as the LDG method in (Cockburn et al., 2008), have the gradient as an independent primal variable. With these methods, for diffusion or diffusion-dominated problems, the primal gradient and primal scalar variable fields can be used to postprocess a scalar field that converges with order in the norm, where is the polynomial order of the primal scalar variable. However, as advection becomes dominant, the higher order convergence is lost and consequently so is the value of having the gradient as an independent variable. In advection-dominated cases, interior penalty HDG methods, such as that outlined in (Rhebergen and Wells, 2017), may be a good choice.

Implementation in MOOSE

HDG kernels derive from Kernels. However, the methods that must be overridden are quite different. These are onElement and onInternalSide, which implement integrations in the volume of elements and on internal faces respectively. External boundary condition integration occurs in HDG Boundary Conditions.

Within onElement and onInternalSide, hybridized kernel developers have eight different data structures they need to populate. Six are inherited from the HDGData class). These are


  /// Matrix data structures for on-diagonal coupling
  EigenMatrix _PrimalMat, _LMMat;
  /// Vector data structures
  EigenVector _PrimalVec, _LMVec;
  /// Matrix data structures for off-diagonal coupling
  EigenMatrix _PrimalLM, _LMPrimal;
```
And the two declared in `HDGKernel`:
```
  /// Containers for the global degree of freedom numbers for primal and LM variables
  /// respectively
  std::vector<dof_id_type> _primal_dof_indices;
  std::vector<dof_id_type> _lm_dof_indices;

The _PrimalMat holds the Jacobian entries for the dependence of primal degrees of freedom on primal degrees of freedom; _LMMat is dependence of LM dofs on LM dofs; _PrimalLM is dependence of primal dofs on LM dofs; _LMPrimal is dependence of LM dofs on primal dofs. The _PrimalVec and _LMVec objects hold the residuals for the primal and LM degrees of freedom respectively. _primal_dof_indices and _lm_dof_indices hold the primal and LM global degree of freedom numbers respectively for the current element. HDGIntegratedBC classes also inherit from HDGData and must also fill the six matrix and vector structures within their onBoundary method.

Note that local finite element assembly occurs twice within a single iteration of Newton's method. The first assembly occurs prior to the linear solve and adds into the global residual and Jacobian data structures which represent only the trace/Lagrange-multiplier degrees of freedom. The linear solve then occurs which computes the Newton update for the Lagrange multiplier degrees of freedom. This Lagrange multiplier increment is then used in the second assembly post-linear-solve to compute the primal variable solution increment. Because only the Lagrange multiplier variables and their degrees of freedom participate in the global solve, they are the only variables that live in the nonlinear system. The primal variables live in the auxiliary system.

References

  1. Tan Bui-Thanh. From godunov to a unified hybridized discontinuous galerkin framework for partial differential equations. Journal of Computational Physics, 295:114–146, 2015.[BibTeX]
  2. Bernardo Cockburn, Bo Dong, and Johnny Guzmán. A superconvergent ldg-hybridizable galerkin method for second-order elliptic problems. Mathematics of Computation, 77(264):1887–1916, 2008.[BibTeX]
  3. Bernardo Cockburn, Jayadeep Gopalakrishnan, and Raytcho Lazarov. Unified hybridization of discontinuous galerkin, mixed, and continuous galerkin methods for second order elliptic problems. SIAM Journal on Numerical Analysis, 47(2):1319–1365, 2009.[BibTeX]
  4. Ngoc Cuong Nguyen, Jaime Peraire, and Bernardo Cockburn. A hybridizable discontinuous galerkin method for stokes flow. Computer Methods in Applied Mechanics and Engineering, 199(9-12):582–597, 2010.[BibTeX]
  5. Ngoc Cuong Nguyen, Jaume Peraire, and Bernardo Cockburn. An implicit high-order hybridizable discontinuous galerkin method for the incompressible navier–stokes equations. Journal of Computational Physics, 230(4):1147–1170, 2011.[BibTeX]
  6. Sander Rhebergen and Garth N Wells. Analysis of a hybridized/interface stabilized finite element method for the stokes equations. SIAM Journal on Numerical Analysis, 55(4):1982–2003, 2017.[BibTeX]