Overview

LehrFEM++ includes built-in lf::assemble::EntityMatrixProvider (EMPs) and lf::assemble::EntityVectorProvider (EVPs) for common PDEs. Custom EMPs/EVPs can also be defined. These providers compute local element matrices/vectors for finite element methods and are typically passed to lf::assemble::AssembleMatrixLocally or lf::assemble::AssembleVectorLocally (see Assembly).

Custom Entity Vector Providers

A custom EMP must implement the lf::assemble::EntityMatrixProvider concept. Minimal definitions:

class ElemMatProvider {
 public:
  // Constructor can be used to pass data required for local computations
  ElemMatProvider() = default;
  // Select cells taken into account during cell-oriented assembly
  bool isActive(const lf::mesh::Entity& cell);
  // Compute element matrix for a cell, here a fixed-size matrix
  Eigen::Matrix<double, 3, 3> Eval(const lf::mesh::Entity& cell);
};

class ElemVecProvider {
 public:
  // Constructor can be used to pass data required for local computations
  ElemVecProvider() = default;
  // Select cells taken into account during cell-oriented assembly
  bool isActive(const lf::mesh::Entity& cell);
  // Compute element vector for a cell
  Eigen::Vector3d Eval(const lf::mesh::Entity& cell);
};

Built-in Entity Matrix Providers

Below is a summary of key built-in EMPs:

Equation	Name (follow link for details)	Description
\( \int\limits_{K}\boldsymbol{\alpha}(\mathbf{x})\mathbf{grad}\,u \cdot\mathbf{grad}\,v\,\mathrm{d}\mathbf{x} \)	fe::DiffusionElementMatrixProvider	stiffness matrix for the diffusion equation.
\( \int\limits_{K}\gamma(\mathbf{x})u\,\overline{v}\,\mathrm{d}\mathbf{x} \)	fe::MassElementMatrixProvider	mass matrix for the reaction-diffusion equation.
\( \int\limits_e\gamma(x)u(x)\overline{v(x)}\,\mathrm{d}S(x) \)	fe::MassEdgeMatrixProvider	mass matrix for the reaction-diffusion equation.
\( \int\limits_{K}\mathbf{grad}\,u\cdot\mathbf{grad}\,v\,\mathrm{d}\mathbf{x} \)	uscalfe::LinearFELaplaceElementMatrix	stiffness matrix for the Laplace equation.
\( \int\limits_{K}\boldsymbol{\alpha}(\mathbf{x})\mathbf{grad}\,u\cdot\mathbf{grad}\,v\,\mathrm{d}\mathbf{x} \) \( + \int\limits_{K}\gamma(\mathbf{x})u\,\overline{v}\,\mathrm{d}\mathbf{x} \)	uscalfe::ReactionDiffusionElementMatrixProvider	stiffness matrix for the reaction-diffusion equation.
\( \int\limits_e\gamma(x)u(x)\overline{v(x)}\,\mathrm{d}S(x) \)	uscale::MassEdgeMatrixProvider	mass matrix for the reaction-diffusion equation.

lf::fe::DiffusionElementMatrixProvider

\[ (u,v) \mapsto\int\limits_{K}\boldsymbol{\alpha}(\mathbf{x})\mathbf{grad}\,u \cdot\mathbf{grad}\,v\,\mathrm{d}\mathbf{x} \; \]

with diffusion coefficient \(\mathbf{\alpha}\), see also Example Lecture Document Example 2.8.3.29

lf::fe::MassElementMatrixProvider

The element matrix corresponds to the (local) bilinear form

\[ (u,v) \mapsto\int\limits_{K}\gamma(\mathbf{x})u\,\overline{v}\,\mathrm{d}\mathbf{x} \;, \]

with reaction coefficient \(\gamma\), see also Lecture Document Example 2.8.3.29

lf::fe::MassEdgeMatrixProvider

The edge matrix corresponds to the (local) bilinear form

\[ (u,v) \mapsto \int\limits_e\gamma(x)u(x)\overline{v(x)}\,\mathrm{d}S(x)\;, \]

where \(e\) is an edge of the mesh, and \(\gamma\) a scalar-valued coefficient function.

lf::uscalfe::LinearFELaplaceElementMatrix

The element matrix corresponds to the (local) bilinear form

\[ (u,v) \mapsto\int\limits_{K}\mathbf{grad}\,u\cdot\mathbf{grad}\,v\,\mathrm{d}\mathbf{x} \;, \]

lf::uscalfe::ReactionDiffusionElementMatrixProvider

\[ (u,v) \mapsto\int\limits_{K}\boldsymbol{\alpha}(\mathbf{x})\mathbf{grad}\,u\cdot\mathbf{grad}\,v + \gamma(\mathbf{x})u\,\overline{v}\,\mathrm{d}\mathbf{x} \;, \]

with diffusion coefficient \(\mathbf{\alpha}\) and reaction coefficient \(\gamma\). See also Lecture Document Example 2.8.3.29.

lf::uscalfe::MassEdgeMatrixProvider

\[ (u,v) \mapsto \int\limits_e\gamma(x)u(x)\overline{v(x)}\,\mathrm{d}S(x)\;, \]

where \(e\) is an edge of the mesh, and \(\gamma\) a scalar-valued coefficient function. See Lecture Document Example 2.7.4.33 for a similar example on assembly of boundary contributions.

Built-in Entity Vector Providers

Entity Vector Providers (EVPs) in LehrFEM++ are used to compute local element vectors for finite element methods. Below is a table summarizing the key built-in EVPs in LehrFEM++.

Equation	Name (follow link for details)	Description
\( \int\limits_{K}f(\mathbf{x})v\,\mathrm{d}\mathbf{x} \)	fe::ScalarLoadElementVectorProvider	load vector for the scalar load on an element.
\( \int\limits_{e}g(x)\overline{v(x)}\,\mathrm{d}S(x) \)	fe::ScalarLoadEdgeVectorProvider	load vector for the scalar load on an edge \(e\).
\( \int\limits_{K}f(\mathbf{x})\overline{v(x)}\,\mathrm{d}\mathbf{x} \)	uscalfe::ScalarLoadElementVectorProvider	load vector for scalar finite elements; volume contributions only.
\( \int\limits_{e}g(x)\overline{v(x)}\,\mathrm{d}S(x) \)	uscalfe::ScalarLoadEdgeVectorProvider	load vector for scalar finite elements
\( \int\limits_{K}f(\mathbf{x})\overline{v(x)}\,\mathrm{d}\mathbf{x} \)	uscalfe::LinearFELocalLoadVector	load vector for scalar linear finite elements.

lf::fe::ScalarLoadElementVectorProvider

lf::fe::ScalarLoadElementVectorProvider computes the local element vector corresponding to the (local) linear form

\[ v \mapsto\int\limits_{K}f(\mathbf{x})v\,\mathrm{d}\mathbf{x} \;, \]

where \(f\) is a scalar-valued function.

lf::fe::ScalarLoadEdgeVectorProvider

lf::fe::ScalarLoadEdgeVectorProvider computes the local edge contributions corresponding to the (local) linear form

\[ v \mapsto\int\limits_{e}g(x)\overline{v(x)}\,\mathrm{d}S(x) \;, \]

where \(e\) is an edge of the mesh, and \(g\) a scalar-valued function.

lf::uscalfe::ScalarLoadElementVectorProvider

lf::uscalfe::ScalarLoadElementVectorProvider computes the local element vector corresponding to the (local) linear form

\[ v \mapsto\int\limits_{K}f(\mathbf{x})\overline{v(x)}\,\mathrm{d}\mathbf{x} \;, \]

where \(f\) is a scalar-valued function. Note: This provider only computes the volume contributions on \(K\).

lf::uscalfe::ScalarLoadEdgeVectorProvider

lf::uscalfe::ScalarLoadEdgeVectorProvider computes the local edge contributions corresponding to the (local) linear form