Quick Reference - Finite Element Spaces

Table of Contents

• Overview
• (General) Scalar Finite Element Space (lf::fe)
  • Hierarchic Scalar Finite Element Space
• Uniform Finite Element Space (lf::uscalfe)
  • Lagrangian Finite Element Spaces

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Attention

The contents of this page are discussed in Lecture Document Paragraph 2.8.3.28 and Lecture Document Section 2.6. Please read before using quick reference.

Overview

LehrFEM++ provides an interface for (scalar) finite element spaces. Scalar means that the approximation space is always scalar-valued, and the shape functions are scalar-valued.

(General) Scalar Finite Element Space (lf::fe)

The lf::fe namespace provides a general interface for scalar finite element spaces. Implementation and usage of scalar FE spaces are covered in more detail in Lecture Document Paragraph 2.8.3.28.

Hierarchic Scalar Finite Element Space

A special case of the general scalar finite element space is a Hierarchic Scalar Finite Element Space, implemented by lf::fe::HierarchicScalarFESpace. Hierarchic FE spaces assign a polynomial degree for the shape functions to each mesh entity. They can easily be constructed from a mesh and a function mapping mesh entities to polynomial degrees.

// generate a simple test mesh
std::shared_ptr<lf::mesh::Mesh> mesh_p =
    lf::mesh::test_utils::GenerateHybrid2DTestMesh(1);
 
// define a polynomial degree function
auto degree = [](const lf::mesh::Entity &entity) -> unsigned {
  return 2;  // all entities have degree 2 -> equivalent to O2 Lagrangian
             // FESpace
};
 
// init HierarchicScalarFESpace on mesh_p
lf::fe::HierarchicScalarFESpace<double> fe_space(mesh_p, degree);

Uniform Finite Element Space (lf::uscalfe)

The lf::uscalfe (Uniform Scalar Finite Elements) namespace is a specialization of lf::fe for uniform scalar finite element spaces. Uniform in this context means that the approximation space has a uniform order of approximation over the whole mesh. In other words, the shape functions of a given approximation space depend only on the underlying reference element of a mesh entity.

Lagrangian Finite Element Spaces

A prominent example of a uniform finite element space is the n-th order Lagrangian finite element spaces. Lecture Document Section 2.6 discusses these spaces and the mathematical background in detail.

LehrFEM++ provides convenience classes for constructing order 1, 2, and 3 Lagrangian finite element spaces:

• lf::uscalfe::FeSpaceLagrangeO1: (Bi)Linear Lagrangian Finite Element space.
• lf::uscalfe::FeSpaceLagrangeO2: Quadratic Lagrangian Finite Element space.
• lf::uscalfe::FeSpaceLagrangeO3: Cubic Lagrangian Finite Element space.

// generate a simple test mesh
std::shared_ptr<lf::mesh::Mesh> mesh_p =
    lf::mesh::test_utils::GenerateHybrid2DTestMesh(1);
 
// init O1 Lagrangian finite element space on mesh_p
lf::uscalfe::FeSpaceLagrangeO1<double> fe_space(mesh_p);