Lecture 2 — Function spaces, meshes, and affine scaling

Lecture 1 introduced the variational formulation of the Poisson problem and the Ritz–Galerkin approximation. The next question is more basic:

What does it mean for a function to be an admissible solution, and how can we systematically build finite-dimensional spaces that approximate it?

This lecture develops the analytical and geometrical language needed for that construction. We introduce Lebesgue and Sobolev spaces, explain weak derivatives, describe triangulations and shape regularity, and derive the scaling rules induced by affine maps from a reference element. These rules are the mechanism behind the interpolation estimates used later in the course.

The guiding principle is that a finite element calculation separates into two layers:

1. an analytic layer, which measures integrability and differentiability in $L^p$ and Sobolev norms;

2. a geometric layer, which transfers functions and polynomial bases between a fixed reference element and every physical cell of the mesh.

The constants in the second layer are harmless only when the mesh is shape-regular. This is why mesh geometry is part of the approximation theory, not merely an implementation detail.

1. Why function spaces enter finite element methods¶

Let $\Omega\subset\mathbb{R}^d$ be an open, bounded domain. A classical formulation of a boundary-value problem asks for a function $u$ satisfying a differential equation at every point of $\Omega$. This formulation is often too restrictive:

• the data $f$ may be integrable without being continuous;

• the exact solution may have derivatives that are not classical functions;

• the boundary of $\Omega$ may be only piecewise smooth;

• a finite element approximation is usually only piecewise polynomial and its higher derivatives may jump across element interfaces.

Instead of asking for pointwise differentiability, we measure the function and its derivatives in integral norms. The weak formulation from Lecture 1 is an example: for Poisson’s equation, it is enough to control first derivatives in $L^2(\Omega)$, which leads naturally to $H^1(\Omega)$ and $H_0^1(\Omega)$.

Throughout this lecture, functions that agree almost everywhere are identified. The spaces therefore contain equivalence classes of functions, rather than a preferred pointwise representative.

2. Multi-indices and weak derivatives¶

A multi-index is a vector $\alpha=(\alpha_1,\ldots,\alpha_d)\in\mathbb{N}_0^d$. We use the notation

For a smooth function, this is the usual derivative. For a function that is only locally integrable, the same notation can be defined in the sense of distributions.

Let $u\in L^1_{\mathrm{loc}}(\Omega)$. A function $v\in L^1_{\mathrm{loc}}(\Omega)$ is the weak derivative $D^\alpha u$ if

The test function $\varphi$ is smooth and compactly supported, so no boundary term appears. If a classical derivative exists and is locally integrable, it is also the weak derivative. Conversely, the weak derivative extends the notion of derivative to functions that are not classically differentiable everywhere.

2.1 Two one-dimensional examples¶

Consider $u(x)=|x|$ on $(-1,1)$. The function is not differentiable at the origin, but it has the weak derivative

Since $\operatorname{sign}(x)\in L^p(-1,1)$ for every finite $p$, we have $u\in W^{1,p}(-1,1)$. The second distributional derivative is $2\delta_0$, which is not an $L^p$-function. Therefore $u\notin W^{2,p}(-1,1)$.

By contrast, a jump is already too singular for $W^{1,p}$. Let $H(x)=0$ for $x<0$ and $H(x)=1$ for $x>0$. Its distributional derivative is $\delta_0$, so $H\notin W^{1,p}(-1,1)$ for any finite $p$. This is the analytic reason why a globally $H^1$-conforming finite element function must be continuous across an interior edge in two dimensions.

3. Lebesgue spaces¶

For $1\leq p<\infty$, the Lebesgue space $L^p(\Omega)$ is

with norm

For $p=\infty$, we use the essential supremum:

The adjective essential means that changing $u$ on a set of measure zero does not change the norm.

If $1\leq p\leq\infty$ and $q$ is its conjugate exponent,

then Hölder’s inequality states

For $p=q=2$, this is the Cauchy–Schwarz inequality. Minkowski’s inequality

makes $L^p(\Omega)$ a Banach space. The special space $L^2(\Omega)$ is a Hilbert space with inner product

For complex-valued functions, the second factor is conjugated. We use real functions in this course unless stated otherwise.

3.1 A useful integrability test¶

Near the origin in $\mathbb{R}^d$, consider $u(x)=|x|^{-\beta}$. In polar coordinates,

Consequently,

This elementary calculation is a useful reminder that integrability depends on both the strength of a singularity and the dimension of the domain.

4. Sobolev spaces¶

For an integer $k\geq0$ and $1\leq p\leq\infty$, define

where the derivatives are weak derivatives. For $p<\infty$, the standard norm is

For $p=\infty$, replace the sum by a maximum:

The order-$k$ seminorm keeps only the derivatives of exactly order $k$:

For $p=\infty$, use the maximum over $|\alpha|=k$. The seminorm may vanish for nonzero functions: if $\Omega$ is connected, then $|u|_{W^{k,p}(\Omega)}=0$ implies that $u$ is a polynomial of degree at most $k-1$.

The Hilbert-space case is denoted by

Its inner product and norm are

We often write $\|u\|_{k,p,\Omega}$ and $|u|_{k,p,\Omega}$ for the norm and seminorm, and omit $\Omega$ when it is clear from the context.

4.1 The spaces $W_0^{k,p}$ and $H_0^1$¶

The space with homogeneous boundary values is defined by closure:

In particular, $H_0^1(\Omega)=W_0^{1,2}(\Omega)$. On a bounded Lipschitz domain, the trace theorem identifies $H_0^1(\Omega)$ with the $H^1$-functions whose trace vanishes on $\partial\Omega$. In the weak Poisson problem of Lecture 1, this is the correct space for both the solution and the test functions.

Poincaré’s inequality states that there is a constant $C_\Omega>0$ such that

Thus, on $H_0^1(\Omega)$, the seminorm

is a norm equivalent to the full $H^1$-norm. This is exactly what allows the Poisson bilinear form to be coercive in Lecture 1.

4.2 Regularity and continuity¶

Sobolev regularity is not the same as pointwise smoothness. A useful rule of thumb is that $W^{k,p}$-functions become continuous when $kp>d$, although the precise statement involves the Sobolev embedding theorems and the regularity of the domain. In two dimensions:

• $H^1(\Omega)=W^{1,2}(\Omega)$ is generally not contained in $C^0(\overline{\Omega})$;

• $W^{1,p}(\Omega)\hookrightarrow C^0(\overline{\Omega})$ for $p>2$;

• $H^2(\Omega)\hookrightarrow C^0(\overline{\Omega})$.

This distinction matters for finite elements. A piecewise polynomial function can be smooth inside each cell and still fail to belong to $H^1(\Omega)$ if it has a jump across an interior face.

5. Triangulations and mesh geometry¶

A mesh is a finite collection of cells that covers the computational domain. For a simplicial mesh in two dimensions, the cells are triangles; in three dimensions, they are tetrahedra. The word triangulation is used more generally here for a partition into simple cells such as triangles, quadrilaterals, tetrahedra, or hexahedra.

We write

where $\mathcal{T}_h$ is the triangulation and $T$ denotes an individual cell. Distinct cells may intersect only in a common lower-dimensional entity:

• the empty set;

• a common vertex;

• a common edge in two dimensions;

• a common face in three dimensions.

The partition must not contain gaps or overlaps in its interior. A mesh may approximate a curved domain by straight-sided cells, or it may use curved geometry through an appropriate mapping. The annular domain used in Lecture 1 is a useful example: the mesh has an inner and an outer boundary, and the finite element calculation must know which faces belong to each boundary.

5.1 Local and global length scales¶

For a cell $T$, let $h_T$ denote its diameter:

For a simplex, let $\rho_T$ denote its inradius, the radius of the largest ball contained in $T$. For a general cell, one may use an analogous inscribed radius. The global mesh size and minimum inradius are

A family of meshes is shape-regular if there is a constant $\sigma>0$, independent of the refinement level, such that

Equivalently, no cell is allowed to become arbitrarily thin relative to its diameter. The constant $\sigma$ controls the constants in the scaling estimates below.

A mesh is quasi-uniform when all its cells have comparable size, for example

Shape regularity is compatible with local refinement: neighbouring cells may have different sizes, as long as their shapes do not degenerate. Quasi-uniformity is stronger and is often convenient for global estimates, but is not necessary for adaptive finite element methods.

5.2 Broken norms¶

If a function is smooth only on each cell, we use a broken Sobolev seminorm:

The corresponding broken norm is defined by summing all derivatives of order at most $k$ cell by cell. A broken norm does not control jumps across faces. To obtain a conforming subspace of $H^1(\Omega)$, the local polynomial pieces must be glued with matching traces on every interior face.

6. The reference element and affine maps¶

The central implementation idea of finite elements is to define shape functions once on a reference cell and transport them to every physical cell.

For a $d$-simplex, use the reference simplex

where $e_1,\ldots,e_d$ are the canonical basis vectors. For a triangle, $\widehat T$ has vertices $(0,0)$, $(1,0)$, and $(0,1)$.

Let $T$ be a physical simplex with vertices $x_0,\ldots,x_d$. The affine map $F_T:\widehat T\to T$ is

If the vertices are not degenerate, $B_T$ is invertible and

The absolute Jacobian determinant is constant on $T$:

The change-of-variables formula is therefore particularly simple:

We use a hat for a function on the reference element:

For first derivatives, the chain rule gives

The matrix $B_T^{-T}$, not $B_T^{-1}$ itself, transforms gradients. This transpose is easy to miss and is important when implementing local stiffness matrices.

7. Scaling of $L^p$ and Sobolev norms¶

The change of variables already gives the exact $L^p$ scaling. If $\widehat v=v\circ F_T$, then, for $1\leq p<\infty$,

For $p=\infty$, the Jacobian does not appear:

For a first derivative, the gradient transformation and the change of variables give

and, in the other direction,

More generally, for integer $k\geq0$, affine changes of variables yield the estimates

and

The hidden constants depend on $k$, $p$, the dimension, and the reference element, but not on the particular physical cell. The estimates use the fact that the derivative of an affine map is the constant matrix $B_T$. For a non-affine map, higher derivatives of the map also enter the chain rule.

7.1 Shape-regular scaling¶

For a shape-regular family of $d$-dimensional simplices,

The last relation uses shape regularity. Without it, the determinant can be much smaller than $h_T^d$ because a cell can be long and thin.

Combining these estimates gives the familiar rule

and the reverse inequality

The exponent has a clear interpretation:

• $d/p$ comes from the volume scaling of the $L^p$-integral;

• $-k$ comes from differentiating $k$ times;

• the geometry is summarized by the single length scale $h_T$ only because the mesh is shape-regular.

For $h_T\leq1$, the lower-order terms can be absorbed into the highest-order term, giving analogous full-norm estimates:

8. Two sanity checks¶

8.1 An interval¶

Let $T=[a,a+h]$ and $\widehat T=[0,1]$. The affine map is

If $\widehat v=v\circ F_T$, then

Equivalently,

For $k=0$, this is the volume scaling $h^{1/p}$. Every derivative adds a factor $h^{-1}$, as expected from the chain rule.

8.2 The Poisson energy on one cell¶

For $p=2$ and $k=1$, the Dirichlet energy on one cell is

up to constants controlled by shape regularity. Therefore:

• in $d=1$, the local stiffness contribution scales like $h_T^{-1}$;

• in $d=2$, it is scale-invariant up to the shape of the cell;

• in $d=3$, it scales like $h_T$.

The local mass integral instead scales as

These two rules already explain important differences between mass matrices and stiffness matrices and are useful when checking an assembly routine.

9. What fails without shape regularity?¶

Consider a family of triangles whose diameter is approximately $h_T$, but whose altitude is $\varepsilon h_T$ with $\varepsilon\to0$. Then

The inverse-gradient estimate contains $\|B_T^{-1}\|$, so its constant grows like $\varepsilon^{-1}$. A formula that depends only on $h_T$ is no longer uniform. In practice, badly shaped cells can lead to:

• large interpolation constants;

• inaccurate numerical integration;

• poorly conditioned stiffness matrices;

• slower or unreliable iterative solver convergence.

Shape regularity is therefore the geometric hypothesis that lets us replace a matrix estimate by a scalar estimate involving $h_T$.

10. From scaling to finite element approximation¶

Let $P^k(\widehat T)$ be a polynomial space on the reference element. The corresponding physical-cell space is

The reference element has finitely many basis functions. Once these are known, the affine map transports them to every cell. This has three consequences:

1. locality: element matrices and vectors can be computed cell by cell;

2. reuse: the reference basis and quadrature rule are shared by all cells;

3. scaling: estimates on $\widehat T$ transfer to $T$ with constants controlled by $h_T$ and shape regularity.

The global space is assembled by identifying local degrees of freedom on shared entities. For a continuous Lagrange space, neighbouring polynomial pieces must have matching values on their common face. This produces a conforming space

and makes Céa’s lemma applicable. If the pieces are allowed to jump, we obtain a broken or discontinuous space instead; that will be the subject of the later DG lectures.

The next analytical step is to quantify how well $P^k(T)$ approximates a smooth function. The Bramble–Hilbert lemma does exactly this on the reference element, while the affine scaling estimates transfer the result to every cell. This is the bridge from the abstract estimate in Lecture 1 to the concrete $h$-dependent error bounds of Lecture 3.

11. Worked checklist¶

For a new mesh and a new finite element implementation, the following checklist captures the logic of this lecture:

1. Identify the domain $\Omega$ and the regularity required of its boundary.

2. Choose the Sobolev space in which the weak solution is sought.

3. Specify the cells $T\in\mathcal{T}_h$, their diameters $h_T$, and their inradii $\rho_T$.

4. Check whether the mesh family is shape-regular and, if needed, quasi-uniform.

5. Choose a reference cell $\widehat T$ and write the affine map $F_T$.

6. Compute $B_T$, $B_T^{-1}$, and $J_T=|\det B_T|$.

7. Transform functions, gradients, and integrals to $\widehat T$.

8. Use the scaling rules to track the powers of $h_T$.

9. Glue local degrees of freedom only when the desired global Sobolev conformity is satisfied.

Summary¶

• $L^p$-spaces measure integrability; $W^{k,p}$-spaces also control weak derivatives up to order $k$.

• $H^k=W^{k,2}$ is a Hilbert space, and $H_0^1$ is the natural energy space for Poisson’s equation with homogeneous Dirichlet data.

• A triangulation partitions the domain into cells, while shape regularity prevents cells from becoming arbitrarily thin.

• Every physical simplex is an affine image of a reference simplex.

• The Jacobian controls volume scaling, while $B_T$ and $B_T^{-1}$ control derivative scaling.

• The factor $h_T^{d/p-k}$ is the fundamental Sobolev scaling law on a shape-regular mesh.

• These estimates allow reference-element calculations to become uniform finite element estimates on the physical mesh.

For further reading, see the discussions of Sobolev spaces and affine scaling in Brenner & Scott (2010) and Ern & Guermond (2004), and the finite element framework in Ciarlet (1978).