Rodin/Variational/P1/QuadratureRule.h file

Optimized quadrature rules for P1 finite element spaces.

This file provides specialized quadrature rules for P1 spaces that exploit the constant gradient property of P1 basis functions to achieve exact integration with minimal quadrature points.

Quadrature Strategies

  • Centroid quadrature: Uses single point at element barycenter
  • Exact for P1 bilinear forms: $ \int \nabla \phi_i \cdot \nabla \phi_j $
  • Reduced integration: Enables efficient assembly

Supported Integrands

  • Scalar P1 shape functions: $ \int v \, dx $
  • Dot products: $ \int f \cdot v \, dx $
  • Mass forms: $ \int (Au) \cdot v \, dx $
  • Stiffness forms: $ \int \nabla u \cdot \nabla v \, dx $
  • Anisotropic stiffness: $ \int (A\nabla u) \cdot \nabla v \, dx $
  • Jacobian forms: $ \int \mathbf{J}u : \mathbf{J}v \, dx $
  • Potential operators for boundary elements

These specializations mirror the generic quadrature rules in Rodin/Variational/QuadratureRule.h but collapse to a single centroid evaluation when possible. Mixed trial/test P1 spaces are supported for the mass, stiffness, and Jacobian forms.

Namespaces

namespace Rodin
The Rodin library for finite element methods and shape optimization.
namespace Rodin::Variational
Namespace Rodin::Variational.

Classes

template<class NestedDerived, class Range, class Mesh>
class Rodin::Variational::QuadratureRule<ShapeFunctionBase<ShapeFunction<NestedDerived, P1<Range, Mesh>, TestSpace>, P1<Range, Mesh>, TestSpace>>
Integration of a P1 ShapeFunction.
template<class LHSDerived, class RHSDerived, class Range, class Mesh>
class Rodin::Variational::QuadratureRule<ShapeFunctionBase<Dot<FunctionBase<LHSDerived>, ShapeFunctionBase<ShapeFunction<RHSDerived, P1<Range, Mesh>, TestSpace>, P1<Range, Mesh>, TestSpace>>, P1<Range, Mesh>, TestSpace>>
Integration of the Dot product of some coefficient function and a P1 ShapeFunction.
template<class LHSDerived, class RHSDerived, class LHSRange, class RHSRange, class LHSMesh, class RHSMesh>
class Rodin::Variational::QuadratureRule<Dot<ShapeFunctionBase<ShapeFunction<LHSDerived, P1<LHSRange, LHSMesh>, TrialSpace>, P1<LHSRange, LHSMesh>, TrialSpace>, ShapeFunctionBase<ShapeFunction<RHSDerived, P1<RHSRange, RHSMesh>, TestSpace>, P1<RHSRange, RHSMesh>, TestSpace>>>
Integration of the isotropic Dot product of two instances of the P1 ShapeFunction.
template<class CoefficientDerived, class LHSDerived, class RHSDerived, class LHSRange, class RHSRange, class LHSMesh, class RHSMesh>
class Rodin::Variational::QuadratureRule<Dot<ShapeFunctionBase<Mult<FunctionBase<CoefficientDerived>, ShapeFunctionBase<ShapeFunction<LHSDerived, P1<LHSRange, LHSMesh>, TrialSpace>, P1<LHSRange, LHSMesh>, TrialSpace>>, P1<LHSRange, LHSMesh>, TrialSpace>, ShapeFunctionBase<ShapeFunction<RHSDerived, P1<RHSRange, RHSMesh>, TestSpace>, P1<RHSRange, RHSMesh>, TestSpace>>>
Integration of the anisotropic Dot product of two instances of the P1 ShapeFunction.
template<class LHSDerived, class RHSDerived, class LHSRange, class RHSRange, class LHSMesh, class RHSMesh>
class Rodin::Variational::QuadratureRule<Dot<ShapeFunctionBase<Grad<ShapeFunction<LHSDerived, P1<LHSRange, LHSMesh>, TrialSpace>>, P1<LHSRange, LHSMesh>, TrialSpace>, ShapeFunctionBase<Grad<ShapeFunction<RHSDerived, P1<RHSRange, RHSMesh>, TestSpace>>, P1<RHSRange, RHSMesh>, TestSpace>>>
Integration of the isotropic Dot product of two instances of the P1 Grad of ShapeFunction.
template<class CoefficientDerived, class LHSDerived, class RHSDerived, class LHSRange, class RHSRange, class LHSMesh, class RHSMesh>
class Rodin::Variational::QuadratureRule<Dot<ShapeFunctionBase<Mult<FunctionBase<CoefficientDerived>, ShapeFunctionBase<Grad<ShapeFunction<LHSDerived, P1<LHSRange, LHSMesh>, TrialSpace>>, P1<LHSRange, LHSMesh>, TrialSpace>>, P1<LHSRange, LHSMesh>, TrialSpace>, ShapeFunctionBase<Grad<ShapeFunction<RHSDerived, P1<RHSRange, RHSMesh>, TestSpace>>, P1<RHSRange, RHSMesh>, TestSpace>>>
Integration of the anisotropic Dot product of two instances of the P1 Grad of ShapeFunction.
template<class CoefficientDerived, class LHSDerived, class RHSDerived, class LHSRange, class RHSRange, class LHSMesh, class RHSMesh>
class Rodin::Variational::QuadratureRule<Mult<FunctionBase<CoefficientDerived>, Dot<ShapeFunctionBase<ShapeFunction<LHSDerived, P1<LHSRange, LHSMesh>, TrialSpace>, P1<LHSRange, LHSMesh>, TrialSpace>, ShapeFunctionBase<ShapeFunction<RHSDerived, P1<RHSRange, RHSMesh>, TestSpace>, P1<RHSRange, RHSMesh>, TestSpace>>>>
Specialization for $\int c \, (u \cdot v) \ dx$ in the case of P1 shape functions.
template<class LHSDerived, class RHSDerived, class LHSMesh, class RHSMesh>
class Rodin::Variational::QuadratureRule<Dot<ShapeFunctionBase<Div<ShapeFunction<LHSDerived, P1<Math::SpatialVector<Real>, LHSMesh>, TrialSpace>>, P1<Math::SpatialVector<Real>, LHSMesh>, TrialSpace>, ShapeFunctionBase<ShapeFunction<RHSDerived, P1<Real, RHSMesh>, TestSpace>, P1<Real, RHSMesh>, TestSpace>>>
Specialization for $\int (\nabla \cdot u)\, q \ dx$ in the case of P1 shape functions.
template<class LHSDerived, class RHSDerived, class LHSMesh, class RHSMesh>
class Rodin::Variational::QuadratureRule<Dot<ShapeFunctionBase<ShapeFunction<LHSDerived, P1<Real, LHSMesh>, TrialSpace>, P1<Real, LHSMesh>, TrialSpace>, ShapeFunctionBase<Div<ShapeFunction<RHSDerived, P1<Math::SpatialVector<Real>, RHSMesh>, TestSpace>>, P1<Math::SpatialVector<Real>, RHSMesh>, TestSpace>>>
Specialization for $\int p \, (\nabla \cdot v)\, dx$ in the case of P1 shape functions.
template<class LHSDerived, class RHSDerived, class LHSRange, class RHSRange, class LHSMesh, class RHSMesh>
class Rodin::Variational::QuadratureRule<Dot<ShapeFunctionBase<Jacobian<ShapeFunction<LHSDerived, P1<LHSRange, LHSMesh>, TrialSpace>>, P1<LHSRange, LHSMesh>, TrialSpace>, ShapeFunctionBase<Jacobian<ShapeFunction<RHSDerived, P1<RHSRange, RHSMesh>, TestSpace>>, P1<RHSRange, RHSMesh>, TestSpace>>>
Integration of the isotropic Frobenius inner product two instances of the P1 Jacobian of ShapeFunction.
template<class CoeffDerived, class LHSDerived, class RHSDerived, class LHSRange, class RHSRange, class LHSMesh, class RHSMesh>
class Rodin::Variational::QuadratureRule<Dot<ShapeFunctionBase<Dot<FunctionBase<CoeffDerived>, ShapeFunctionBase<Jacobian<ShapeFunction<LHSDerived, P1<LHSRange, LHSMesh>, TrialSpace>>, P1<LHSRange, LHSMesh>, TrialSpace>>, P1<LHSRange, LHSMesh>, TrialSpace>, ShapeFunctionBase<Dot<FunctionBase<CoeffDerived>, ShapeFunctionBase<Jacobian<ShapeFunction<RHSDerived, P1<RHSRange, RHSMesh>, TestSpace>>, P1<RHSRange, RHSMesh>, TestSpace>>, P1<RHSRange, RHSMesh>, TestSpace>>>
Integration of a rank-one form in the P1 Jacobian of ShapeFunction.
template<class CoefficientDerived, class LHSDerived, class RHSDerived, class LHSRange, class RHSRange, class LHSMesh, class RHSMesh>
class Rodin::Variational::QuadratureRule<Dot<ShapeFunctionBase<Mult<FunctionBase<CoefficientDerived>, ShapeFunctionBase<Jacobian<ShapeFunction<LHSDerived, P1<LHSRange, LHSMesh>, TrialSpace>>, P1<LHSRange, LHSMesh>, TrialSpace>>, P1<LHSRange, LHSMesh>, TrialSpace>, ShapeFunctionBase<Jacobian<ShapeFunction<RHSDerived, P1<RHSRange, RHSMesh>, TestSpace>>, P1<RHSRange, RHSMesh>, TestSpace>>>
Integration of the anisotropic Frobenius inner product two instances of the P1 Jacobian of ShapeFunction.
template<class CoefficientDerived, class LHSDerived, class RHSDerived, class LHSRange, class RHSRange, class LHSMesh, class RHSMesh>
class Rodin::Variational::QuadratureRule<Dot<ShapeFunctionBase<Mult<ShapeFunctionBase<Jacobian<ShapeFunction<LHSDerived, P1<LHSRange, LHSMesh>, TrialSpace>>, P1<LHSRange, LHSMesh>, TrialSpace>, FunctionBase<CoefficientDerived>>, P1<LHSRange, LHSMesh>, TrialSpace>, ShapeFunctionBase<ShapeFunction<RHSDerived, P1<RHSRange, RHSMesh>, TestSpace>, P1<RHSRange, RHSMesh>, TestSpace>>>
Specialization for $ \int (\nabla u \cdot f) \cdot v $ in the case of P1 shape functions.
template<class Kernel, class Range, class Mesh, class LHSDerived, class RHSDerived>
class Rodin::Variational::QuadratureRule<Dot<Potential<Kernel, ShapeFunctionBase<LHSDerived, P1<Range, Mesh>, TrialSpace>>, ShapeFunctionBase<RHSDerived, P1<Range, Mesh>, TestSpace>>>
Specialization for $\int \mathcal{K}(u) \cdot v$ in the case of P1 trial/test shape functions.