Models

The following models are currently implemented in OpenCMP. In general, all models are expected to include a standard Galerkin finite element method formulation, a discontinuous Galerkin formulation, and a diffuse interface formulation. Nonlinear models are further expected to include formulations for Oseen-style linearization and IMEX linearization. The following sections provide the weak forms for all implemented formulations of the available models.

Poisson Equation

The Poisson equation takes the following form on a simulation domain \Omega. There are three possible types of boundary condition - Dirichlet, Neumann, and Robin - applied on boundary regions \Gamma_D, \Gamma_N, and \Gamma_R respectively.

\eta \nabla^2 u + f &= 0 \mbox{ in } \Omega \\ u &= g \mbox{ on } \Gamma_D \\ -\eta \bm{n} \cdot \bm{\nabla} u &= h \mbox{ on } \Gamma_N \\ \eta \bm{n} \cdot \bm{\nabla} u &= r(u - q) \mbox{ on } \Gamma_R

where \eta is a constant typically taken to be the diffusivity and f is some source function.

The different formulations of the Poisson equation finite element weak form are given below. Note that in all cases v is the trial function. In the case of the discontinuous Galerkin method, \{\} and [[]] refer to the average and jump operators respectively and \alpha is the penalty coefficient. In the case of the diffuse interface method, \phi is the phase field, \{ \phi_{const} \} are masks for different boundary condition regions, and \beta is the Nitsche method penalty parameter. Also note that for the diffuse interface method all integrals become volume integrals over the enclosing simple domain \kappa.

Standard Galerkin Finite Element Formulation

The standard Galerkin finite element method formulation is as follows:

\int_{\Omega} \bm{\nabla} u \cdot \bm{\nabla} v \: dx + \int_{\Gamma_N} \frac{1}{\eta} hv \: ds - \int_{\Gamma_R} \frac{1}{\eta} vr \left( u - q \right) \: ds = \int_{\Omega} \frac{1}{\eta} fv \: dx

Dirichlet boundary conditions are imposed strongly by setting values at applicable boundary degrees of freedom.

Discontinuous Galerkin Formulation

The discontinuous Galerkin formulation is as follows:

&\sum_{\mathcal{K} \in \mathcal{T}} \int_{\mathcal{K}} \bm{\nabla} u \cdot \bm{\nabla} v \: dx \\ - &\sum_{\mathcal{F} \in \mathcal{F}_I} \int_{\mathcal{F}} \left( [[u\bm{n}]] \cdot \{ \bm{\nabla} v \} + [[v\bm{n}]] \cdot \{ \bm{\nabla} u \} - \alpha [[u]] [[v]] \right) \: ds \\ - &\sum_{\mathcal{F} \in \mathcal{F}_D} \int_{\mathcal{F}} \left( \bm{n} \cdot \left( u - g \right) \bm{\nabla} v + \bm{n} \cdot v \bm{\nabla} u - \alpha \left( u - g \right) v \right) \: ds \\ - &\sum_{\mathcal{F} \in \mathcal{F}_N} \int_{\mathcal{F}} \frac{1}{\eta} vh \: ds + \sum_{\mathcal{F} \in \mathcal{F}_R} \int_{\mathcal{F}} \frac{1}{\eta} vr\left( u - q \right) \: ds \\ = &\sum_{\mathcal{K} \in \mathcal{T}} \int_{\mathcal{K}} \frac{1}{\eta} fv \: dx

where volume integrals are over individual mesh elements \mathcal{K} then summed over the entire triangulation \mathcal{T}. Surface integrals are over mesh element facets \mathcal{F} and summed over either interior facets \mathcal{F}_I or boundary facets \mathcal{F}_D, \mathcal{F}_N, and \mathcal{F}_R.

Diffuse Interface Formulation

The diffuse interface formulation is as follows for the standard Galerkin finite element method:

&\int_{\kappa} \bm{\nabla} u \cdot \bm{\nabla} v \phi \: dx + \int_{\kappa} \frac{1}{\eta} hv \lvert \bm{\nabla} \phi \rvert \phi_N \: dx - \int_{\kappa} \frac{1}{\eta} vr \left( u - q \right) \lvert \bm{\nabla} \phi \rvert \phi_R \: dx \\ + &\int_{\kappa} \left( u - g \right) \bm{\nabla} \phi \cdot \bm{\nabla} v \phi_D \: dx + \int_{\kappa} v \bm{\nabla} \phi \cdot \bm{\nabla} u \phi_D \: dx + \beta \int_{\kappa} v \left( u - g \right) \lvert \bm{\nabla} \phi \rvert \phi_D \: dx \\ = &\int_{\kappa} \frac{1}{\eta} fv \phi \: dx

and for the discontinuous Galerkin method:

&\sum_{\mathcal{K} \in \mathcal{T}} \int_{\mathcal{K}} \bm{\nabla} u \cdot \bm{\nabla} v \phi \: dx \\ - &\sum_{\mathcal{F} \in \mathcal{F}_I} \int_{\mathcal{F}} \left( [[u\bm{n}]] \cdot \{ \bm{\nabla} v \} + [[v\bm{n}]] \cdot \{ \bm{\nabla} u \} - \alpha [[u]] [[v]] \right) \phi \: ds \\ + &\sum_{\mathcal{K} \in \mathcal{T}} \int_{\mathcal{K}} \left( \left( u - g \right) \bm{\nabla} \phi \cdot \bm{\nabla} v + v \bm{\nabla} \phi \cdot \bm{\nabla} u + \beta \left( u - g \right) v \lvert \bm{\nabla} \phi \rvert \right) \phi_D \: dx \\ + &\sum_{\mathcal{K} \in \mathcal{T}} \int_{\mathcal{K}} \frac{1}{\eta} vh \lvert \bm{\nabla} \phi \rvert \phi_N \: dx + \sum_{\mathcal{K} \in \mathcal{T}} \int_{\mathcal{K}} \frac{1}{\eta} vr\left( u - q \right) \lvert \bm{\nabla} \phi \rvert \phi_R \: dx \\ = &\sum_{\mathcal{K} \in \mathcal{T}} \int_{\mathcal{K}} \frac{1}{\eta} fv \phi \: dx

Stokes Equations

The Stokes equations take the following form on a simulation domain \Omega. There are two possible types of boundary condition - velocity Dirichlet and normal stress - applied on boundary regions \Gamma_D and \Gamma_S respectively.

-\nu \nabla^2 \bm{u} + \bm{\nabla} p &= \bm{f} \mbox{ in } \Omega \\ \bm{\nabla} \cdot \bm{u} &= 0 \mbox{ in } \Omega \\ \bm{u} &= \bm{g} \mbox{ on } \Gamma_D \\ \bm{n} \cdot \left(-\nu \bm{\nabla} \bm{u} + p \mathbb{I} \right) &= \bm{h} \mbox{ on } \Gamma_S

where \bm{u} is the velocity and p is the pressure. Furthermore, \nu is the constant kinematic viscosity and f is some body force.

The different formulations of the Stokes equations finite element weak form are given below. Note that in all cases \bm{v} and q are the trial functions for velocity and pressure respectively. In the case of the discontinuous Galerkin method, \{\} and [[]] refer to the average and jump operators respectively and \alpha is the penalty coefficient. In the case of the diffuse interface method, \phi is the phase field, \{ \phi_{const} \} are masks for different boundary condition regions, and \beta is the Nitsche method penalty parameter. Also note that for the diffuse interface method all integrals become volume integrals over the enclosing simple domain \kappa.

Standard Galerkin Finite Element Formulation

The standard Galerkin finite element method formulation is as follows:

\int_{\Omega} \left( \nu \bm{\nabla} \bm{u} : \bm{\nabla} \bm{v} - p \left( \bm{\nabla} \cdot \bm{v} \right) - q \left( \bm{\nabla} \cdot \bm{u} \right) \right) \: dx + \int_{\Gamma_S} \bm{v} \cdot \bm{h} \: ds &= \int_{\Omega} \bm{v} \cdot \bm{f} \: dx

Dirichlet boundary conditions are imposed strongly by setting values at applicable boundary degrees of freedom.

Discontinuous Galerkin Formulation

The discontinuous Galerkin formulation is as follows:

&\sum_{\mathcal{K} \in \mathcal{T}} \int_{\mathcal{K}} \left( \nu \bm{\nabla} \bm{u} : \bm{\nabla} \bm{v} - p \left( \bm{\nabla} \cdot \bm{v} \right) - q \left( \bm{\nabla} \cdot \bm{u} \right) \right) \: dx \\ - &\sum_{\mathcal{F} \in \mathcal{F}_I} \int_{\mathcal{F}} \nu \left( [[\bm{u}\bm{n}]] : \{ \bm{\nabla} \bm{v} \} + [[\bm{v}\bm{n}]] : \{ \bm{\nabla} \bm{u} \} - \alpha [[\bm{u} \bm{n}]] : [[\bm{v} \bm{n}]] \right) \: ds \\ - &\sum_{\mathcal{F} \in \mathcal{F}_D} \int_{\mathcal{F}} \nu \left( \left( \bm{u} - \bm{g} \right) \bm{n} : \bm{\nabla} \bm{v} + \bm{v} \bm{n} : \bm{\nabla} \bm{u} - \alpha \left( \bm{u} - \bm{g} \right) \cdot \bm{v} \right) \: ds \\ - &\sum_{\mathcal{F} \in \mathcal{F}_S} \int_{\mathcal{F}} \bm{v} \cdot \bm{h} \: ds \\ = &\sum_{\mathcal{K} \in \mathcal{T}} \int_{\mathcal{K}} \bm{v} \cdot \bm{f} \: dx

where volume integrals are over individual mesh elements \mathcal{K} then summed over the entire triangulation \mathcal{T}. Surface integrals are over mesh element facets \mathcal{F} and summed over either interior facets \mathcal{F}_I or boundary facets \mathcal{F}_D and \mathcal{F}_S.

Diffuse Interface Formulation

The diffuse interface formulation is as follows for the standard Galerkin finite element method:

&\int_{\kappa} \left( \nu \bm{\nabla} \bm{u} : \bm{\nabla} \bm{v} - p \left( \bm{\nabla} \cdot \bm{v} \right) - q \left( \bm{\nabla} \cdot \bm{u} \right) \right) \phi \: dx + \int_{\kappa} \bm{v} \cdot \bm{h} \lvert \bm{\nabla} \phi \rvert \phi_S \: dx \\ + &\int_{\kappa} \left( \bm{u} - \bm{g} \right) \cdot \bm{\nabla} \bm{v} \cdot \bm{\nabla} \phi \phi_D \: dx + \int_{\kappa} \bm{v} \cdot \bm{\nabla} \bm{u} \cdot \bm{\nabla} \phi \phi_D \: dx + \beta \int_{\kappa} \bm{v} \cdot \left( \bm{u} - \bm{g} \right) \lvert \bm{\nabla} \phi \rvert \phi_D \: dx \\ = &\int_{\kappa} \bm{v} \cdot \bm{f} \phi \: dx

and for the discontinuous Galerkin method:

&\sum_{\mathcal{K} \in \mathcal{T}} \int_{\mathcal{K}} \left( \nu \bm{\nabla} \bm{u} : \bm{\nabla} \bm{v} - p \left( \bm{\nabla} \cdot \bm{v} \right) - q \left( \bm{\nabla} \cdot \bm{u} \right) \right) \phi \: dx \\ - &\sum_{\mathcal{F} \in \mathcal{F}_I} \int_{\mathcal{F}} \nu \left( [[\bm{u}\bm{n}]] : \{ \bm{\nabla} \bm{v} \} + [[\bm{v}\bm{n}]] : \{ \bm{\nabla} \bm{u} \} - \alpha [[\bm{u} \bm{n}]] : [[\bm{v} \bm{n}]] \right) \phi \: ds \\ + &\sum_{\mathcal{K} \in \mathcal{T}} \int_{\mathcal{K}} \nu \left( \left( \bm{u} - \bm{g} \right) \bm{\nabla} \phi : \bm{\nabla} \bm{v} + \bm{v} \bm{\nabla} \phi : \bm{\nabla} \bm{u} - \beta \left( \bm{u} - \bm{g} \right) \cdot \bm{v} \lvert \bm{\nabla} \phi \rvert \right) \phi_D \: dx \\ - &\sum_{\mathcal{K} \in \mathcal{T}} \int_{\mathcal{K}} \bm{v} \cdot \bm{h} \lvert \bm{\nabla} \phi \rvert \phi_S \: dx \\ = &\sum_{\mathcal{K} \in \mathcal{T}} \int_{\mathcal{K}} \bm{v} \cdot \bm{f} \phi \: dx

Incompressible Navier-Stokes Equations

The incompressible Navier-Stokes equations take the following form on a simulation domain \Omega. There are two possible types of boundary condition - velocity Dirichlet and normal stress - applied on boundary regions \Gamma_D and \Gamma_S respectively.

\frac{\partial \bm{u}}{\partial t} + \bm{\nabla} \cdot \left( \bm{u} \bm{u} \right) - \nu \nabla^2 \bm{u} + \bm{\nabla} p &= \bm{f} \mbox{ in } \Omega \\ \bm{\nabla} \cdot \bm{u} &= 0 \mbox{ in } \Omega \\ \bm{u}(t=0) &= \bm{u}_0 \mbox{ in } \Omega \\ p(t=0) &= p_0 \mbox{ in } \Omega \\ \bm{u} &= \bm{g} \mbox{ on } \Gamma_D \\ \bm{n} \cdot \left(\bm{u} \bm{u} - \nu \bm{\nabla} \bm{u} + p \mathbb{I} \right) - \max \left( \bm{u} \cdot \bm{n},0 \right) \bm{u} &= \bm{h} \mbox{ on } \Gamma_S

where \bm{u} is the velocity and p is the pressure. Furthermore, \nu is the constant kinematic viscosity and f is some body force. When Oseen-style linearization is used to linearize the nonlinear convection term, one velocity in said term with be replaced by a known velocity field \bm{w} (usually the velocity from the previous time step).

The different formulations of the incompressible Navier-Stokes equations finite element weak form are given below. Note that in all cases \bm{v} and q are the trial functions for velocity and pressure respectively. In the case of the discontinuous Galerkin method, \{\} and [[]] refer to the average and jump operators respectively and \alpha is the penalty coefficient. In the case of the diffuse interface method, \phi is the phase field, \{ \phi_{const} \} are masks for different boundary condition regions, and \beta is the Nitsche method penalty parameter. Also note that for the diffuse interface method all integrals become volume integrals over the enclosing simple domain \kappa.

Standard Galerkin Finite Element Formulation

The standard Galerkin finite element method formulation is as follows for Oseen-style linearization:

&\int_{\Omega} \left( \bm{v} \cdot \frac{\partial \bm{u}}{\partial t} - \bm{u} \bm{w} : \bm{\nabla} \bm{v} + \nu \bm{\nabla} \bm{u} : \bm{\nabla} \bm{v} - p \left( \bm{\nabla} \cdot \bm{v} \right) - q \left( \bm{\nabla} \cdot \bm{u} \right) \right) \: dx \\ + &\int_{\Gamma_S} \bm{v} \cdot \left( \bm{h} + \max \left( \bm{w} \cdot \bm{n}, 0 \right) \bm{u} \right) \: ds \\ = &\int_{\Omega} \bm{v} \cdot \bm{f} \: dx

and IMEX time discretization:

&\int_{\Omega} \left( \bm{v} \cdot \frac{\partial \bm{u}}{\partial t} + \nu \bm{\nabla} \bm{u} : \bm{\nabla} \bm{v} - p \left( \bm{\nabla} \cdot \bm{v} \right) - q \left( \bm{\nabla} \cdot \bm{u} \right) \right) \: dx \\ + &\int_{\Gamma_S} \bm{v} \cdot \left( \bm{h} + \max \left( \bm{w} \cdot \bm{n},0 \right) \bm{u} \right) \: ds \\ = &\int_{\Omega} \left( \bm{v} \cdot \bm{f} - \bm{u} \cdot \bm{\nabla} \bm{u} \cdot \bm{v} \right) \: dx

In both cases, Dirichlet boundary conditions are imposed strongly by setting values at applicable boundary degrees of freedom.

Discontinuous Galerkin Formulation

The discontinuous Galerkin formulation is as follows for Oseen-style linearization:

&\sum_{\mathcal{K} \in \mathcal{T}} \int_{\mathcal{K}} \left( \bm{v} \cdot \frac{\partial \bm{u}}{\partial t} - \bm{u} \bm{w} : \bm{\nabla} \bm{v} + \nu \bm{\nabla} \bm{u} : \bm{\nabla} \bm{v} - p \left( \bm{\nabla} \cdot \bm{v} \right) - q \left( \bm{\nabla} \cdot \bm{u} \right) \right) \: dx \\ + &\sum_{\mathcal{F} \in \mathcal{F}_I} \int_{\mathcal{F}} [[\bm{v} \bm{n}]] : \left( \{ \bm{u} \} \left( \bm{w} \cdot \bm{n} \right) \bm{n} + \frac{1}{2} \left( \bm{u}^+ - \bm{u}^- \right) \lvert \bm{w} \cdot \bm{n} \rvert \bm{n} \right) \: ds \\ - &\sum_{\mathcal{F} \in \mathcal{F}_I} \int_{\mathcal{F}} \nu \left( [[\bm{u}\bm{n}]] : \{ \bm{\nabla} \bm{v} \} + [[\bm{v}\bm{n}]] : \{ \bm{\nabla} \bm{u} \} - \alpha [[\bm{u} \bm{n}]] : [[\bm{v} \bm{n}]] \right) \: ds \\ + &\sum_{\mathcal{F} \in \mathcal{F}_D} \int_{\mathcal{F}} \bm{v} \bm{n} : \left( \frac{1}{2} \left( \bm{u} + \bm{g} \right) \left( \bm{w} \cdot \bm{n} \right) \bm{n} + \frac{1}{2} \left( \bm{u} - \bm{g} \right) \lvert \bm{w} \cdot \bm{n} \rvert \bm{n} \right) \: ds \\ - &\sum_{\mathcal{F} \in \mathcal{F}_D} \int_{\mathcal{F}} \nu \left( \left( \bm{u} - \bm{g} \right) \bm{n} : \bm{\nabla} \bm{v} + \bm{v} \bm{n} : \bm{\nabla} \bm{u} - \alpha \left( \bm{u} - \bm{g} \right) \cdot \bm{v} \right) \: ds \\ + &\sum_{\mathcal{F} \in \mathcal{F}_S} \int_{\mathcal{F}} \bm{v} \cdot \left( \bm{h} + \max \left( \bm{w} \cdot \bm{n}, 0 \right) \bm{u} \right) \: ds \\ = &\sum_{\mathcal{K} \in \mathcal{T}} \int_{\mathcal{K}} \bm{v} \cdot \bm{f} \: dx

and IMEX time discretization:

&\sum_{\mathcal{K} \in \mathcal{T}} \int_{\mathcal{K}} \left( \bm{v} \cdot \frac{\partial \bm{u}}{\partial t} + \nu \bm{\nabla} \bm{u} : \bm{\nabla} \bm{v} - p \left( \bm{\nabla} \cdot \bm{v} \right) - q \left( \bm{\nabla} \cdot \bm{u} \right) \right) \: dx \\ - &\sum_{\mathcal{F} \in \mathcal{F}_I} \int_{\mathcal{F}} \nu \left( [[\bm{u}\bm{n}]] : \{ \bm{\nabla} \bm{v} \} + [[\bm{v}\bm{n}]] : \{ \bm{\nabla} \bm{u} \} - \alpha [[\bm{u} \bm{n}]] : [[\bm{v} \bm{n}]] \right) \: ds \\ - &\sum_{\mathcal{F} \in \mathcal{F}_D} \int_{\mathcal{F}} \nu \left( \left( \bm{u} - \bm{g} \right) \bm{n} : \bm{\nabla} \bm{v} + \bm{v} \bm{n} : \bm{\nabla} \bm{u} - \alpha \left( \bm{u} - \bm{g} \right) \cdot \bm{v} \right) \: ds \\ + &\sum_{\mathcal{F} \in \mathcal{F}_S} \int_{\mathcal{F}} \bm{v} \cdot \left( \bm{h} + \max \left( \bm{w} \cdot \bm{n}, 0 \right) \bm{u} \right) \: ds \\ = &\sum_{\mathcal{K} \in \mathcal{T}} \int_{\mathcal{K}} \left( \bm{v} \cdot \bm{f} - \bm{u} \cdot \bm{\nabla} \bm{u} \cdot \bm{v} \right) \: dx

In both cases, volume integrals are over individual mesh elements \mathcal{K} then summed over the entire triangulation \mathcal{T}. Surface integrals are over mesh element facets \mathcal{F} and summed over either interior facets \mathcal{F}_I or boundary facets \mathcal{F}_D and \mathcal{F}_S.

Diffuse Interface Formulation

The diffuse interface formulation is as follows for the standard Galerkin finite element method using Oseen-style linearization:

&\int_{\kappa} \left( \bm{v} \cdot \frac{\partial \bm{u}}{\partial t} - \bm{u} \bm{w} : \bm{\nabla} \bm{v} + \nu \bm{\nabla} \bm{u} : \bm{\nabla} \bm{v} - p \left( \bm{\nabla} \cdot \bm{v} \right) - q \left( \bm{\nabla} \cdot \bm{u} \right) \right) \phi \: dx \\ + &\int_{\kappa} \bm{v} \cdot \left( \bm{h} + \max \left( \bm{w} \cdot \bm{n}, 0 \right) \bm{u} \right) \lvert \bm{\nabla} \phi \rvert \phi_S \: dx \\ + &\int_{\kappa} \left( \bm{u} - \bm{g} \right) \cdot \bm{\nabla} \bm{v} \cdot \bm{\nabla} \phi \phi_D \: dx + \int_{\kappa} \bm{v} \cdot \bm{\nabla} \bm{u} \cdot \bm{\nabla} \phi \phi_D \: dx + \beta \int_{\kappa} \bm{v} \cdot \left( \bm{u} - \bm{g} \right) \lvert \bm{\nabla} \phi \rvert \phi_D \: dx \\ = &\int_{\Omega} \bm{v} \cdot \bm{f} \: dx

and IMEX time discretization:

&\int_{\kappa} \left( \bm{v} \cdot \frac{\partial \bm{u}}{\partial t} + \nu \bm{\nabla} \bm{u} : \bm{\nabla} \bm{v} - p \left( \bm{\nabla} \cdot \bm{v} \right) - q \left( \bm{\nabla} \cdot \bm{u} \right) \right) \phi \: dx \\ + &\int_{\kappa} \bm{v} \cdot \left( \bm{h} + \max \left( \bm{w} \cdot \bm{n},0 \right) \bm{u} \right) \lvert \bm{\nabla} \phi \rvert \phi_S \: dx \\ + &\int_{\kappa} \left( \bm{u} - \bm{g} \right) \cdot \bm{\nabla} \bm{v} \cdot \bm{\nabla} \phi \phi_D \: dx + \int_{\kappa} \bm{v} \cdot \bm{\nabla} \bm{u} \cdot \bm{\nabla} \phi \phi_D \: dx + \beta \int_{\kappa} \bm{v} \cdot \left( \bm{u} - \bm{g} \right) \lvert \bm{\nabla} \phi \rvert \phi_D \: dx \\ = &\int_{\kappa} \left( \bm{v} \cdot \bm{f} - \bm{u} \cdot \bm{\nabla} \bm{u} \cdot \bm{v} \right) \phi \: dx

The diffuse interface formulation is as follows for the discontinuous Galerkin method using Oseen-style linearization:

&\sum_{\mathcal{K} \in \mathcal{T}} \int_{\mathcal{K}} \left( \bm{v} \cdot \frac{\partial \bm{u}}{\partial t} - \bm{u} \bm{w} : \bm{\nabla} \bm{v} + \nu \bm{\nabla} \bm{u} : \bm{\nabla} \bm{v} - p \left( \bm{\nabla} \cdot \bm{v} \right) - q \left( \bm{\nabla} \cdot \bm{u} \right) \right) \phi \: dx \\ + &\sum_{\mathcal{F} \in \mathcal{F}_I} \int_{\mathcal{F}} [[\bm{v} \bm{n}]] : \left( \{ \bm{u} \} \left( \bm{w} \cdot \bm{n} \right) \bm{n} + \frac{1}{2} \left( \bm{u}^+ - \bm{u}^- \right) \lvert \bm{w} \cdot \bm{n} \rvert \bm{n} \right) \: ds \\ - &\sum_{\mathcal{F} \in \mathcal{F}_I} \int_{\mathcal{F}} \nu \left( [[\bm{u}\bm{n}]] : \{ \bm{\nabla} \bm{v} \} + [[\bm{v}\bm{n}]] : \{ \bm{\nabla} \bm{u} \} - \alpha [[\bm{u} \bm{n}]] : [[\bm{v} \bm{n}]] \right) \: ds \\ - &\sum_{\mathcal{K} \in \mathcal{T}} \int_{\mathcal{K}} \bm{v} \cdot \left( \frac{1}{2} \left( \bm{u} + \bm{g} \right) \left( \bm{w} \cdot \bm{\nabla} \phi \right) + \frac{1}{2} \left( \bm{u} - \bm{g} \right) \lvert \bm{w} \cdot \bm{\nabla} \phi \rvert \right) \phi_D \: dx \\ + &\sum_{\mathcal{K} \in \mathcal{T}} \int_{\mathcal{K}} \nu \left( \left( \bm{u} - \bm{g} \right) \bm{\nabla} \phi : \bm{\nabla} \bm{v} + \bm{v} \bm{\nabla} \phi : \bm{\nabla} \bm{u} - \alpha \left( \bm{u} - \bm{g} \right) \cdot \bm{v} \lvert \bm{\nabla} \phi \rvert \right) \phi_D \: dx \\ + &\sum_{\mathcal{K} \in \mathcal{T}} \int_{\mathcal{K}} \bm{v} \cdot \left( \bm{h} + \max \left( \bm{w} \cdot \bm{n}, 0 \right) \bm{u} \right) \lvert \bm{\nabla} \phi \rvert \phi_D \: dx \\ = &\sum_{\mathcal{K} \in \mathcal{T}} \int_{\mathcal{K}} \bm{v} \cdot \bm{f} \phi \: dx

and IMEX time discretization:

&\sum_{\mathcal{K} \in \mathcal{T}} \int_{\mathcal{K}} \left( \bm{v} \cdot \frac{\partial \bm{u}}{\partial t} + \nu \bm{\nabla} \bm{u} : \bm{\nabla} \bm{v} - p \left( \bm{\nabla} \cdot \bm{v} \right) - q \left( \bm{\nabla} \cdot \bm{u} \right) \right) \phi \: dx \\ - &\sum_{\mathcal{F} \in \mathcal{F}_I} \int_{\mathcal{F}} \nu \left( [[\bm{u}\bm{n}]] : \{ \bm{\nabla} \bm{v} \} + [[\bm{v}\bm{n}]] : \{ \bm{\nabla} \bm{u} \} - \alpha [[\bm{u} \bm{n}]] : [[\bm{v} \bm{n}]] \right) \phi \: ds \\ + &\sum_{\mathcal{K} \in \mathcal{T}} \int_{\mathcal{K}} \nu \left( \left( \bm{u} - \bm{g} \right) \bm{\nabla} \phi : \bm{\nabla} \bm{v} + \bm{v} \bm{\nabla} \phi : \bm{\nabla} \bm{u} - \alpha \left( \bm{u} - \bm{g} \right) \cdot \bm{v} \lvert \bm{\nabla} \phi \rvert \right) \phi_D \: dx \\ + &\sum_{\mathcal{K} \in \mathcal{T}} \int_{\mathcal{K}} \bm{v} \cdot \left( \bm{h} + \max \left( \bm{w} \cdot \bm{n}, 0 \right) \bm{u} \right) \lvert \bm{\nabla} \phi \rvert \phi_S \: dx \\ = &\sum_{\mathcal{K} \in \mathcal{T}} \int_{\mathcal{K}} \left( \bm{v} \cdot \bm{f} - \bm{u} \cdot \bm{\nabla} \bm{u} \cdot \bm{v} \right) \phi \: dx

Multi-Component Flow

Note

To be written once discontinuous Galerkin and diffuse interface formulations are available for the multi-component flow model.