InterfaceReaction

Description

Specie M transports between two domains (domain 1 and domain 2), at the interface consider the following reaction is taking place:

$var element = document.getElementById("moose-equation-fbaa681c-d99a-4a63-8fb6-7fde73002581");katex.render("M(1)\\xrightleftharpoons[k_b]{k_f}M(2)", element, {displayMode:true,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$

With the first order reaction rate assuming a quasi-steady-state

$var element = document.getElementById("moose-equation-a497b772-22d7-4b4c-97bd-1f5ed95c17d8");katex.render("Reaction Rate = \\frac {\\partial C_1} {\\partial t} = k_f C_1 - k_b C_2 \\approx 0", element, {displayMode:true,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$

where $var element = document.getElementById("moose-equation-cd53e0aa-0620-4942-be5e-d7198ac0969a");katex.render("C_1", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ is the specie concentration in domain 1, $var element = document.getElementById("moose-equation-75c9aebe-42da-4058-95dc-12e9c2be831a");katex.render("C_2", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ is the specie concentration in domain 2, $var element = document.getElementById("moose-equation-a411cd3e-54e2-45a7-bbd8-2d38fa8e036d");katex.render("k_f", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ is the forward reaction coefficient, and $var element = document.getElementById("moose-equation-150fcbc3-9986-4b68-affc-3172592d7074");katex.render("k_b", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ is the backward reaction coefficient. InterfaceReaction object is used to impose this condition. Associated kernel is:

InterfaceReaction.C

/framework/src/interfacekernels/InterfaceReaction.C

// This file is part of the MOOSE framework
// https://www.mooseframework.org
//
// All rights reserved, see COPYRIGHT for full restrictions
// https://github.com/idaholab/moose/blob/master/COPYRIGHT
//
// Licensed under LGPL 2.1, please see LICENSE for details
// https://www.gnu.org/licenses/lgpl-2.1.html

#include "InterfaceReaction.h"

registerMooseObject("MooseApp", InterfaceReaction);

defineLegacyParams(InterfaceReaction);

InputParameters
InterfaceReaction::validParams()
{
  InputParameters params = InterfaceKernel::validParams();
  params.addRequiredParam<Real>("kf", "Forward reaction rate coefficient.");
  params.addRequiredParam<Real>("kb", "Backward reaction rate coefficient.");
  params.addClassDescription("Implements a reaction to establish ReactionRate=k_f*u-k_b*v "
                             "at interface.");
  return params;
}

InterfaceReaction::InterfaceReaction(const InputParameters & parameters)
  : InterfaceKernel(parameters), _kf(getParam<Real>("kf")), _kb(getParam<Real>("kb"))
{
}

Real
InterfaceReaction::computeQpResidual(Moose::DGResidualType type)
{
  Real r = 0;
  switch (type)
  {
    // Move all the terms to the LHS to get residual, for master domain
    // Residual = kf*u - kb*v = kf*u - kb*v
    // Weak form for master domain is: (test, kf*u - kb*v)
    case Moose::Element:
      r = _test[_i][_qp] * (_kf * _u[_qp] - _kb * _neighbor_value[_qp]);
      break;

    // Similarly, weak form for slave domain is: -(test, kf*u - kb*v),
    // flip the sign because the direction is opposite.
    case Moose::Neighbor:
      r = -_test_neighbor[_i][_qp] * (_kf * _u[_qp] - _kb * _neighbor_value[_qp]);
      break;
  }
  return r;
}

Real
InterfaceReaction::computeQpJacobian(Moose::DGJacobianType type)
{
  Real jac = 0;
  switch (type)
  {
    case Moose::ElementElement:
      jac = _test[_i][_qp] * _kf * _phi[_j][_qp];
      break;
    case Moose::NeighborNeighbor:
      jac = -_test_neighbor[_i][_qp] * -_kb * _phi_neighbor[_j][_qp];
      break;
    case Moose::NeighborElement:
      jac = -_test_neighbor[_i][_qp] * _kf * _phi[_j][_qp];
      break;
    case Moose::ElementNeighbor:
      jac = _test[_i][_qp] * -_kb * _phi_neighbor[_j][_qp];
      break;
  }
  return jac;
}

InterfaceReaction.h

/framework/include/interfacekernels/InterfaceReaction.h

// This file is part of the MOOSE framework
// https://www.mooseframework.org
//
// All rights reserved, see COPYRIGHT for full restrictions
// https://github.com/idaholab/moose/blob/master/COPYRIGHT
//
// Licensed under LGPL 2.1, please see LICENSE for details
// https://www.gnu.org/licenses/lgpl-2.1.html

#pragma once

#include "InterfaceKernel.h"

// Forward Declarations
class InterfaceReaction;

template <>
InputParameters validParams<InterfaceReaction>();

/**
 * Implements a reaction to establish ReactionRate=k_f*u-k_b*v
 * at interface.
 */
class InterfaceReaction : public InterfaceKernel
{
public:
  static InputParameters validParams();

  InterfaceReaction(const InputParameters & parameters);

protected:
  virtual Real computeQpResidual(Moose::DGResidualType type) override;
  virtual Real computeQpJacobian(Moose::DGJacobianType type) override;

  /// Forward reaction rate coefficient
  Real _kf;

  /// Backward reaction rate coefficient
  Real _kb;
};

In addition, fluxes are matched from both domains, this could be achieved by InterfaceDiffusion.

Both kernels at the interface work together to give full mathematical and physical meaning of the problem.

Two examples (steady-state and transient-state) are shown in the MOOSE test directory,

reaction_1D_steady.i

/test/tests/interfacekernels/1d_interface/reaction_1D_steady.i

# Steady-state test for the InterfaceReaction kernel.
#
# Specie M transport from domain 1 (0<=x<=1) to domain 2 (1<x<=2),
# u and v are concentrations in domain 1 and domain 2.
#
# Diffusion in both domains can be described by Ficks law and diffusion
# kernel is applied.
#
# Specie M has different diffusity in different domains, here set as D1=4, D2=2.
#
# Dirichlet boundary conditions are applied, i.e., u(0)=1, v(2)=0
#
# At the interface consider the following
#
# (a) Fluxes are matched from both domains (InterfaceDiffusion kernel)
#
# (b) First-order reaction is R = kf*u - kb*v
#
# Analytical solution is
# u = -0.2*u+1,    0<=u<=1
# v = -0.4*v+0.8,  1<v<=2

[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 1
    nx = 10
    xmax = 2
  []
  [./subdomain1]
    input = gen
    type = SubdomainBoundingBoxGenerator
    bottom_left = '1.0 0 0'
    block_id = 1
    top_right = '2.0 1.0 0'
  [../]
  [./interface]
    type = SideSetsBetweenSubdomainsGenerator
    input = 'subdomain1'
    master_block = '0'
    paired_block = '1'
    new_boundary = 'master0_interface'
  [../]
[]

[Variables]
  [./u]
    order = FIRST
    family = LAGRANGE
    block = '0'
  [../]
  [./v]
    order = FIRST
    family = LAGRANGE
    block = '1'
  [../]
[]

[Kernels]
  [./diff_u]
    type = MatDiffusion
    variable = u
    block = '0'
    diffusivity = D
  [../]
  [./diff_v]
    type = MatDiffusion
    variable = v
    block = '1'
    diffusivity = D
  [../]
[]

[InterfaceKernels]
  [./interface]
    type = InterfaceDiffusion
    variable = u
    neighbor_var = 'v'
    boundary = 'master0_interface'
    D = D
    D_neighbor = D
  [../]
  [./interface_reaction]
    type = InterfaceReaction
    variable = u
    neighbor_var = 'v'
    boundary = 'master0_interface'
    kf = 1 # Forward reaction rate coefficient
    kb = 2 # Backward reaction rate coefficient
  [../]
[]

[BCs]
  [./left]
    type = DirichletBC
    variable = u
    boundary = 'left'
    value = 1
  [../]
  [./right]
    type = DirichletBC
    variable = v
    boundary = 'right'
    value = 0
  [../]
[]

[Materials]
  [./block0]
    type = GenericConstantMaterial
    block = '0'
    prop_names = 'D'
    prop_values = '4'
  [../]
  [./block1]
    type = GenericConstantMaterial
    block = '1'
    prop_names = 'D'
    prop_values = '2'
  [../]
[]

[Preconditioning]
  [./smp]
    type = SMP
    full = true
  [../]
[]

[Executioner]
  type = Steady
  solve_type = PJFNK
  nl_rel_tol = 1e-10
[]

[Outputs]
  print_linear_residuals = true
  execute_on = 'FINAL'
  exodus = true
  csv = true
[]

[Debug]
  show_var_residual_norms = true
[]

[Postprocessors]
  [./elemental_error_u]
    type = ElementL2Error
    function = -0.2*x+1
    variable = 'u'
    block = '0'
  [../]
  [./elemental_error_v]
    type = ElementL2Error
    function = -0.4*x+0.8
    variable = 'v'
    block = '1'
  [../]
[]

reaction_1D_transient.i

/test/tests/interfacekernels/1d_interface/reaction_1D_transient.i

# Transient-state test for the InterfaceReaction kernel.
#
# Same to steady-state, except the following
#
# Natural BCs are applied (i.e. NewmannBC h=0 at left and right)

[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 1
    nx = 10
    xmax = 2
  []
  [./subdomain1]
    input = gen
    type = SubdomainBoundingBoxGenerator
    bottom_left = '1.0 0 0'
    block_id = 1
    top_right = '2.0 1.0 0'
  [../]
  [./interface]
    type = SideSetsBetweenSubdomainsGenerator
    input = 'subdomain1'
    master_block = '0'
    paired_block = '1'
    new_boundary = 'master0_interface'
  [../]
[]

[Variables]
  [./u]
    order = FIRST
    family = LAGRANGE
    block = '0'
  [../]
  [./v]
    order = FIRST
    family = LAGRANGE
    block = '1'
  [../]
[]

[Kernels]
  [./diff_u]
    type = MatDiffusion
    diffusivity = D
    variable = u
    block = '0'
  [../]
  [./diff_v]
    type = MatDiffusion
    diffusivity = D
    variable = v
    block = '1'
  [../]
  [./diff_u_dt]
    type = TimeDerivative
    variable = u
    block = '0'
  [../]
  [./diff_v_dt]
    type = TimeDerivative
    variable = v
    block = '1'
  [../]
  [./source_u]
    type = BodyForce
    variable = u
    block = '0'
  [../]
[]

[InterfaceKernels]
  [./interface]
    type = InterfaceDiffusion
    variable = u
    neighbor_var = 'v'
    boundary = 'master0_interface'
    D = D
    D_neighbor = D
  [../]
  [./interface_reaction]
    type = InterfaceReaction
    variable = u
    neighbor_var = 'v'
    boundary = 'master0_interface'
    kf = 1 # Forward reaction rate coefficient
    kb = 2 # Backward reaction rate coefficient
  [../]
[]

[Materials]
  [./block0]
    type = GenericConstantMaterial
    block = '0'
    prop_names = 'D'
    prop_values = '4'
  [../]
  [./block1]
    type = GenericConstantMaterial
    block = '1'
    prop_names = 'D'
    prop_values = '2'
  [../]
[]

[Preconditioning]
  [./smp]
    type = SMP
    full = true
  [../]
[]

[Executioner]
  type = Transient
  num_steps = 10
  dt = 0.1
  solve_type = NEWTON
[]

[Outputs]
  exodus = true
  print_linear_residuals = true
[]

[Debug]
  show_var_residual_norms = true
[]

Example Input Syntax

  [./interface_reaction]
    type = InterfaceReaction
    variable = u
    neighbor_var = 'v'
    boundary = 'master0_interface'
    kf = 1 # Forward reaction rate coefficient
    kb = 2 # Backward reaction rate coefficient
  [../]

/opt/civet/build_0/moose/test/tests/interfacekernels/1d_interface/reaction_1D_steady.i

# Steady-state test for the InterfaceReaction kernel.
#
# Specie M transport from domain 1 (0<=x<=1) to domain 2 (1<x<=2),
# u and v are concentrations in domain 1 and domain 2.
#
# Diffusion in both domains can be described by Ficks law and diffusion
# kernel is applied.
#
# Specie M has different diffusity in different domains, here set as D1=4, D2=2.
#
# Dirichlet boundary conditions are applied, i.e., u(0)=1, v(2)=0
#
# At the interface consider the following
#
# (a) Fluxes are matched from both domains (InterfaceDiffusion kernel)
#
# (b) First-order reaction is R = kf*u - kb*v
#
# Analytical solution is
# u = -0.2*u+1,    0<=u<=1
# v = -0.4*v+0.8,  1<v<=2

[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 1
    nx = 10
    xmax = 2
  []
  [./subdomain1]
    input = gen
    type = SubdomainBoundingBoxGenerator
    bottom_left = '1.0 0 0'
    block_id = 1
    top_right = '2.0 1.0 0'
  [../]
  [./interface]
    type = SideSetsBetweenSubdomainsGenerator
    input = 'subdomain1'
    master_block = '0'
    paired_block = '1'
    new_boundary = 'master0_interface'
  [../]
[]

[Variables]
  [./u]
    order = FIRST
    family = LAGRANGE
    block = '0'
  [../]
  [./v]
    order = FIRST
    family = LAGRANGE
    block = '1'
  [../]
[]

[Kernels]
  [./diff_u]
    type = MatDiffusion
    variable = u
    block = '0'
    diffusivity = D
  [../]
  [./diff_v]
    type = MatDiffusion
    variable = v
    block = '1'
    diffusivity = D
  [../]
[]

[InterfaceKernels]
  [./interface]
    type = InterfaceDiffusion
    variable = u
    neighbor_var = 'v'
    boundary = 'master0_interface'
    D = D
    D_neighbor = D
  [../]
  [./interface_reaction]
    type = InterfaceReaction
    variable = u
    neighbor_var = 'v'
    boundary = 'master0_interface'
    kf = 1 # Forward reaction rate coefficient
    kb = 2 # Backward reaction rate coefficient
  [../]
[]

[BCs]
  [./left]
    type = DirichletBC
    variable = u
    boundary = 'left'
    value = 1
  [../]
  [./right]
    type = DirichletBC
    variable = v
    boundary = 'right'
    value = 0
  [../]
[]

[Materials]
  [./block0]
    type = GenericConstantMaterial
    block = '0'
    prop_names = 'D'
    prop_values = '4'
  [../]
  [./block1]
    type = GenericConstantMaterial
    block = '1'
    prop_names = 'D'
    prop_values = '2'
  [../]
[]

[Preconditioning]
  [./smp]
    type = SMP
    full = true
  [../]
[]

[Executioner]
  type = Steady
  solve_type = PJFNK
  nl_rel_tol = 1e-10
[]

[Outputs]
  print_linear_residuals = true
  execute_on = 'FINAL'
  exodus = true
  csv = true
[]

[Debug]
  show_var_residual_norms = true
[]

[Postprocessors]
  [./elemental_error_u]
    type = ElementL2Error
    function = -0.2*x+1
    variable = 'u'
    block = '0'
  [../]
  [./elemental_error_v]
    type = ElementL2Error
    function = -0.4*x+0.8
    variable = 'v'
    block = '1'
  [../]
[]

Input Parameters

kbBackward reaction rate coefficient.
C++ Type:double
Options:
Description:Backward reaction rate coefficient.
kfForward reaction rate coefficient.
C++ Type:double
Options:
Description:Forward reaction rate coefficient.
neighbor_varThe variable on the other side of the interface.
C++ Type:std::vector
Options:
Description:The variable on the other side of the interface.
variableThe name of the variable that this boundary condition applies to
C++ Type:NonlinearVariableName
Options:
Description:The name of the variable that this boundary condition applies to

Required Parameters

boundaryThe list of boundary IDs from the mesh where this boundary condition applies
C++ Type:std::vector
Options:
Description:The list of boundary IDs from the mesh where this boundary condition applies
diag_save_inThe name of auxiliary variables to save this Kernel's diagonal Jacobian contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)
C++ Type:std::vector
Options:
Description:The name of auxiliary variables to save this Kernel's diagonal Jacobian contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)
diag_save_in_var_sideThis parameter must exist if diag_save_in variables are specified and must have the same length as diag_save_in. This vector specifies whether the corresponding aux_var should save-in jacobian contributions from the master ('m') or slave side ('s').
C++ Type:MultiMooseEnum
Options:m s
Description:This parameter must exist if diag_save_in variables are specified and must have the same length as diag_save_in. This vector specifies whether the corresponding aux_var should save-in jacobian contributions from the master ('m') or slave side ('s').
save_inThe name of auxiliary variables to save this Kernel's residual contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)
C++ Type:std::vector
Options:
Description:The name of auxiliary variables to save this Kernel's residual contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)
save_in_var_sideThis parameter must exist if save_in variables are specified and must have the same length as save_in. This vector specifies whether the corresponding aux_var should save-in residual contributions from the master ('m') or slave side ('s').
C++ Type:MultiMooseEnum
Options:m s
Description:This parameter must exist if save_in variables are specified and must have the same length as save_in. This vector specifies whether the corresponding aux_var should save-in residual contributions from the master ('m') or slave side ('s').

Optional Parameters

control_tagsAdds user-defined labels for accessing object parameters via control logic.
C++ Type:std::vector
Options:
Description:Adds user-defined labels for accessing object parameters via control logic.
enableTrueSet the enabled status of the MooseObject.
Default:True
C++ Type:bool
Options:
Description:Set the enabled status of the MooseObject.
implicitTrueDetermines whether this object is calculated using an implicit or explicit form
Default:True
C++ Type:bool
Options:
Description:Determines whether this object is calculated using an implicit or explicit form
use_displaced_meshFalseWhether or not this object should use the displaced mesh for computation. Note that in the case this is true but no displacements are provided in the Mesh block the undisplaced mesh will still be used.
Default:False
C++ Type:bool
Options:
Description:Whether or not this object should use the displaced mesh for computation. Note that in the case this is true but no displacements are provided in the Mesh block the undisplaced mesh will still be used.

Advanced Parameters

extra_matrix_tagsThe extra tags for the matrices this Kernel should fill
C++ Type:std::vector
Options:
Description:The extra tags for the matrices this Kernel should fill
extra_vector_tagsThe extra tags for the vectors this Kernel should fill
C++ Type:std::vector
Options:
Description:The extra tags for the vectors this Kernel should fill
matrix_tagssystemThe tag for the matrices this Kernel should fill
Default:system
C++ Type:MultiMooseEnum
Options:nontime system
Description:The tag for the matrices this Kernel should fill
vector_tagsnontimeThe tag for the vectors this Kernel should fill
Default:nontime
C++ Type:MultiMooseEnum
Options:nontime time
Description:The tag for the vectors this Kernel should fill

Tagging Parameters

Input Files

test/tests/interfacekernels/1d_interface/reaction_1D_transient.i
test/tests/interfacekernels/1d_interface/reaction_1D_steady.i

test/tests/interfacekernels/1d_interface/reaction_1D_transient.i

# Transient-state test for the InterfaceReaction kernel.
#
# Same to steady-state, except the following
#
# Natural BCs are applied (i.e. NewmannBC h=0 at left and right)

[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 1
    nx = 10
    xmax = 2
  []
  [./subdomain1]
    input = gen
    type = SubdomainBoundingBoxGenerator
    bottom_left = '1.0 0 0'
    block_id = 1
    top_right = '2.0 1.0 0'
  [../]
  [./interface]
    type = SideSetsBetweenSubdomainsGenerator
    input = 'subdomain1'
    master_block = '0'
    paired_block = '1'
    new_boundary = 'master0_interface'
  [../]
[]

[Variables]
  [./u]
    order = FIRST
    family = LAGRANGE
    block = '0'
  [../]
  [./v]
    order = FIRST
    family = LAGRANGE
    block = '1'
  [../]
[]

[Kernels]
  [./diff_u]
    type = MatDiffusion
    diffusivity = D
    variable = u
    block = '0'
  [../]
  [./diff_v]
    type = MatDiffusion
    diffusivity = D
    variable = v
    block = '1'
  [../]
  [./diff_u_dt]
    type = TimeDerivative
    variable = u
    block = '0'
  [../]
  [./diff_v_dt]
    type = TimeDerivative
    variable = v
    block = '1'
  [../]
  [./source_u]
    type = BodyForce
    variable = u
    block = '0'
  [../]
[]

[InterfaceKernels]
  [./interface]
    type = InterfaceDiffusion
    variable = u
    neighbor_var = 'v'
    boundary = 'master0_interface'
    D = D
    D_neighbor = D
  [../]
  [./interface_reaction]
    type = InterfaceReaction
    variable = u
    neighbor_var = 'v'
    boundary = 'master0_interface'
    kf = 1 # Forward reaction rate coefficient
    kb = 2 # Backward reaction rate coefficient
  [../]
[]

[Materials]
  [./block0]
    type = GenericConstantMaterial
    block = '0'
    prop_names = 'D'
    prop_values = '4'
  [../]
  [./block1]
    type = GenericConstantMaterial
    block = '1'
    prop_names = 'D'
    prop_values = '2'
  [../]
[]

[Preconditioning]
  [./smp]
    type = SMP
    full = true
  [../]
[]

[Executioner]
  type = Transient
  num_steps = 10
  dt = 0.1
  solve_type = NEWTON
[]

[Outputs]
  exodus = true
  print_linear_residuals = true
[]

[Debug]
  show_var_residual_norms = true
[]

test/tests/interfacekernels/1d_interface/reaction_1D_steady.i

# Steady-state test for the InterfaceReaction kernel.
#
# Specie M transport from domain 1 (0<=x<=1) to domain 2 (1<x<=2),
# u and v are concentrations in domain 1 and domain 2.
#
# Diffusion in both domains can be described by Ficks law and diffusion
# kernel is applied.
#
# Specie M has different diffusity in different domains, here set as D1=4, D2=2.
#
# Dirichlet boundary conditions are applied, i.e., u(0)=1, v(2)=0
#
# At the interface consider the following
#
# (a) Fluxes are matched from both domains (InterfaceDiffusion kernel)
#
# (b) First-order reaction is R = kf*u - kb*v
#
# Analytical solution is
# u = -0.2*u+1,    0<=u<=1
# v = -0.4*v+0.8,  1<v<=2

[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 1
    nx = 10
    xmax = 2
  []
  [./subdomain1]
    input = gen
    type = SubdomainBoundingBoxGenerator
    bottom_left = '1.0 0 0'
    block_id = 1
    top_right = '2.0 1.0 0'
  [../]
  [./interface]
    type = SideSetsBetweenSubdomainsGenerator
    input = 'subdomain1'
    master_block = '0'
    paired_block = '1'
    new_boundary = 'master0_interface'
  [../]
[]

[Variables]
  [./u]
    order = FIRST
    family = LAGRANGE
    block = '0'
  [../]
  [./v]
    order = FIRST
    family = LAGRANGE
    block = '1'
  [../]
[]

[Kernels]
  [./diff_u]
    type = MatDiffusion
    variable = u
    block = '0'
    diffusivity = D
  [../]
  [./diff_v]
    type = MatDiffusion
    variable = v
    block = '1'
    diffusivity = D
  [../]
[]

[InterfaceKernels]
  [./interface]
    type = InterfaceDiffusion
    variable = u
    neighbor_var = 'v'
    boundary = 'master0_interface'
    D = D
    D_neighbor = D
  [../]
  [./interface_reaction]
    type = InterfaceReaction
    variable = u
    neighbor_var = 'v'
    boundary = 'master0_interface'
    kf = 1 # Forward reaction rate coefficient
    kb = 2 # Backward reaction rate coefficient
  [../]
[]

[BCs]
  [./left]
    type = DirichletBC
    variable = u
    boundary = 'left'
    value = 1
  [../]
  [./right]
    type = DirichletBC
    variable = v
    boundary = 'right'
    value = 0
  [../]
[]

[Materials]
  [./block0]
    type = GenericConstantMaterial
    block = '0'
    prop_names = 'D'
    prop_values = '4'
  [../]
  [./block1]
    type = GenericConstantMaterial
    block = '1'
    prop_names = 'D'
    prop_values = '2'
  [../]
[]

[Preconditioning]
  [./smp]
    type = SMP
    full = true
  [../]
[]

[Executioner]
  type = Steady
  solve_type = PJFNK
  nl_rel_tol = 1e-10
[]

[Outputs]
  print_linear_residuals = true
  execute_on = 'FINAL'
  exodus = true
  csv = true
[]

[Debug]
  show_var_residual_norms = true
[]

[Postprocessors]
  [./elemental_error_u]
    type = ElementL2Error
    function = -0.2*x+1
    variable = 'u'
    block = '0'
  [../]
  [./elemental_error_v]
    type = ElementL2Error
    function = -0.4*x+0.8
    variable = 'v'
    block = '1'
  [../]
[]

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