Finite element and box-method discretizations for fractional elliptic problems with quadrature and mass lumping

TL;DR

This paper develops a unified framework for finite element and box-method discretizations of fractional elliptic problems, analyzing their error estimates and the role of quadrature and mass lumping.

Contribution

It introduces a novel inner product-based approach that unifies finite element and box-method discretizations for fractional elliptic operators, including new error analysis.

Findings

Mass-lumped inner product yields fractional box discretization.

Error estimates are established for both discretizations.

Numerical experiments confirm theoretical results.

Abstract

We analyze numerical approximation of the fractional elliptic problem $L^{\beta}u=f$, ${\beta>0}$, where $L$ is a second-order self-adjoint elliptic operator with homogeneous Dirichlet or Neumann boundary conditions. The paper develops a unified conforming piecewise linear framework that covers both the standard finite element discretization and the box-method discretization of fractional powers. The key point is that the discrete fractional operator is defined with respect to an admissible inner product on the trial space. This includes, in particular, the standard $L^{2}$ inner product and the quadrature-based mass-lumped inner product, and we also identify a broader family of admissible inner products interpolating between these two realizations. Within this framework, we show that the mass-lumped choice yields the intrinsic fractional box discretization, namely the one obtained by taking fractional powers of the nonfractional box solution operator. For both the finite element and box-method realizations, we establish error estimates under natural consistency assumptions, making explicit the effect of load quadrature in the box case. The analysis applies directly to practical schemes and is supported by numerical experiments in one and two space dimensions.