Constructive Representation of Functions in $N$-Dimensional Sobolev Space

TL;DR

The paper addresses representing functions in the Sobolev space S2^δ[Ω] on a hyperrectangle Ω in terms of their highest-order mixed derivatives and suitably chosen boundary values. It establishes a constructive expansion u(s) = ∑_{0^N ≤ α ≤ δ} (G^δ_α B^{α-δ} D^α u)(s) and proves a bijection between the boundary-data space L2^δ[Ω] and S2^δ[Ω] via the operator 𝒢^δ. Building on this representation, it develops two projection-based approximation schemes: Legendre polynomial projections and step-function projections of the boundary derivatives, coupled with the reconstruction formula, yielding an L2-optimal approach with enhanced Sobolev-norm convergence when higher-order derivatives are projected. Numerical experiments in 1D and 2D demonstrate comparable L2 performance to standard projections while achieving improved Sobolev convergence when projecting higher-order derivatives, confirming the practical viability of the method. The work suggests extensions to more general domains and non-polynomial kernels, broadening the applicability of constructive Sobolev representations.

Abstract

A new representation is proposed for functions in a Sobolev space with dominating mixed smoothness on an $N$-dimensional hyperrectangle. In particular, it is shown that these functions can be expressed in terms of their highest-order mixed derivative, as well as their lower-order derivatives evaluated along suitable boundaries of the domain. The proposed expansion is proven to be invertible, uniquely identifying any function in the Sobolev space with its derivatives and boundary values. Since these boundary values are either finite-dimensional, or exist in the space of square-integrable functions, this offers a bijective relation between the Sobolev space and $L_{2}$. Using this bijection, it is shown how approximation of functions in Sobolev space can be performed in the less restrictive space $L_{2}$, reconstructing such an approximation of the function from an $L_{2}$-optimal projection of its boundary values and highest-order derivative. This approximation method is presented using a basis of Legendre polynomials and a basis of step functions, and results using both bases are demonstrated to exhibit better convergence behavior than a direct projection approach for two numerical examples.

Figures (4)

• Figure 1: $L_{2}$ (left) and Sobolev (right) norms of error in polynomial approximations $P_{d}^{\gamma}u$ of function $u\in S_{2}^{5}[-1,1]$ defined in \ref{['eq:example1']}. The plot $P_{d}u$ corresponds to a standard Legendre projection of $u$ using the $L_{2}$ inner product, as defined in \ref{['eq:projection_leg']}. The plots $P_{d}^{\gamma}u$ correspond to reconstructing an approximation of $u$ from a projection of $\partial_{s}^{\gamma}u$ as in \ref{['eq:projection_leg_sob']}, using Theorem \ref{['thm:fundamental_expansion_ND']}. Projecting the highest-order derivative $\partial_{s}^{5}u$ of $u\in S_{2}^{5}$, both the $L_{2}$ and Sobolev norms of the error in the associated approximation $P_{d}^{(5)}u$ decrease with the degree $d$, with the $L_{2}$ norm decaying at the same rate as observed for the standard projection $P_{d}u=P_{d}^{(0)}u$.
• Figure 2: $L_{2}$ (left) and Sobolev (right) norms of error in step function approximations $Q_{K}^{\gamma}u$ of $u\in S_{2}^{5}[-1,1]$ defined in \ref{['eq:example1']}, on a uniform grid of $K$ cells. Each $Q_{K}^{\gamma}u$ is computed by projecting $D^{\gamma}u$ onto a space spanned by $K$ orthogonal step functions, and subsequently reconstructing an approximation of $u$ as in \ref{['eq:projection_stepfun_sob']}. The error in the direct step function projection $Q_{K}u=Q_{K}^{0}u$ is also plotted.
• Figure 3: $L_{2}$ (left) and Sobolev (right) norms of error in polynomial approximations $P_{d}^{\gamma}w$ of $w\in S_{2}^{(3,3)}[[-1,1]^2]$ defined in \ref{['eq:example2']}. Each approximation $P_{d}^{\gamma}w$ is computed by projecting the derivative $D^{\gamma}w$ onto the space of polynomials of degree at most $d$ in each variable, and reconstructing an approximation of $w$ as in \ref{['eq:projection_leg_sob']}. The error in the direct projection $P_{d}w=P_{d}^{(0,0)}w$ is also plotted. Convergence of the approximation in the Sobolev norm is observed only for $P_{d}^{(2,2)}w$ and $P_{d}^{(3,3)}w$, i.e. when projecting $D^{(2,2)}w$ and $D^{(3,3)}w$.
• Figure 4: $L_{2}$ (left) and Sobolev (right) norms of error in step function approximations $Q_{K}^{\gamma}w$ of $w\in S_{2}^{(3,3)}[[-1,1]^2]$ defined in \ref{['eq:example2']}, on a grid of $K\times K$ cells. Each $Q_{K}^{\gamma}w$ is computed by projecting $D^{\gamma}w$ onto a space spanned by $K\times K$ orthogonal step functions, and subsequently reconstructing an approximation of $w$ as in \ref{['eq:projection_stepfun_sob']}. The error in the direct step function projection $Q_{K}w=Q_{K}^{(0,0)}w$ is also plotted. Note that $Q_{K}^{(3,3)}w=w$ for $K\geq 4$, since $D^{(3,3)}w$ is a piecewise constant function on $4\times 4$ cells.

Theorems & Definitions (10)

• Theorem 1
• Theorem 1
• Lemma 2
• Example 3
• Corollary 4
• Example 5
• Remark 6
• Proposition 7
• Proposition 8
• Corollary 9