Linear Interpolators

$ \newcommand{\cross}{\times} \newcommand{\inner}{\cdot} \newcommand{\div}{\nabla\cdot} \newcommand{\curl}{\nabla\times} \newcommand{\grad}{\nabla} \newcommand{\ddx}[1]{\frac{d#1}{dx}} \newcommand{\abs}[1]{|#1|} $

Linear interpolators can be very useful for interpolating one discrete representation of a field onto another set of basis functions to produce another representation. However, this must be done with care because different discrete representations are not completely interchangeable.

As an example consider a scalar field projected onto either piece-wise linear ($H_1$) or piece-wise constant ($L_2$) basis functions. Interpolating from an $H_1$ representation to an $L_2$ representation should produce a reasonable result because the constant value needed in each element can be computed as a weighted sum of the $H_1$ basis functions in that element. On the other hand, if we try to interpolate from the $L_2$ representation to an $H_1$ representation we don't have enough information to determine reasonable values for the degrees of freedom which are shared between neighboring elements because linear interpolators can only access one element at a time. To accurately compute an $H_1$ representation from an $L_2$ representation requires the type of weighted average of values from neighboring elements that bilinear forms provide but this requires a linear solve and often suitable boundary conditions.

The operators produced by the BilinearForm classes involve integrations and therefore they sum the various contributions from neighboring elements to compute a full integral. The DiscreteLinearOperator classes are not performing integrals but rather interpolations and as such they do not combine contributions from different elements in any way. Consequently if the LinearInterpolators produce different results for entities that are shared between neighboring elements then the resulting representation will depend on the order in which the elements are processed. Such operators are not good candidates for DiscreteLinearOperators. The sections below will offer some guidance on the appropriate use of these operators.

In the tables below the Space column refers to finite element spaces which implement the following methods:

Space Operator Derivative Operator
H1 CalcShape CalcDShape
ND CalcVShape CalcCurlShape
RT CalcVShape CalcDivShape
L2 CalcShape None

The Coef. column refers to the types of coefficients that are available. A boldface coefficient type is required whereas most coefficients are optional.

Coef. Type
S Scalar Valued Function
V Vector Valued Function
D Diagonal Matrix Function
M General Matrix Function

Derivative Interpolators

The $H(Curl)$ and $H(Div)$ spaces are specifically designed to support these derivative operators by having the necessary inter-element continuity. Other possible derivative operators would not possess the correct continuity and must therefore be implemented in a weak sense.

Class Name Domain Range Operator
GradientInterpolator H1 ND $\grad u$
CurlInterpolator ND in 3D RT $\curl\vec{u}$
CurlInterpolator ND in 2D L2 $\hat{z}\cdot(\curl\vec{u})$
DivergenceInterpolator RT L2 $\div\vec{u}$

Product Interpolators

These operators require a bit more care than the previous set. In order for these operators to produce valid results the product of the coefficient with the domain space must be uniquely representable within the desired range space. Additionally, it may sometimes be desirable for the range space to have a higher order than the domain space if the coefficient is not constant. For example if the domain space and the coefficient are both linear it might be desirable, though not necessary, for the range space to be quadratic.

Class Name Domain Range Coef. Operator
ScalarProductInterpolator H1,L2 H1,L2 S $\lambda u$
ScalarVectorProductInterpolator ND,RT ND,RT S $\lambda\vec{u}$
VectorScalarProductInterpolator H1,L2 ND,RT V $\vec{\lambda}u$
VectorCrossProductInterpolator ND,RT in 3D ND,RT V $\vec{\lambda}\times\vec{u}$
ScalarCrossProductInterpolator ND,RT in 2D H1,L2 V $\hat{z}\cdot(\vec{\lambda}\times\vec{u})$
VectorInnerProductInterpolator ND,RT H1,L2 V $\vec{\lambda}\cdot\vec{u}$

Special Purpose Interpolators

Class Name Domain Range Operator
IdentityInterpolator H1,L2 H1,L2 $u$
NormalInterpolator H1$^d$ RT_Trace $\hat{n}\cdot\vec{u}$