arXiv subjects

Kelvin J. R. Almeida-Sousa

Publications and source records attributed to Kelvin J. R. Almeida-Sousa.

3 recordsLinked to original sources

arXiv2026

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

We analyze numerical approximation of the fractional elliptic problem $L^βu=f$, ${β>0}$, 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.

math.NA↗

arXiv2026

Mass Lumping and Numerical Quadrature for Approximation of Fractional Elliptic Differential Equations Driven by Gaussian White Noise

Fractional elliptic stochastic partial differential equations (SPDEs) are widely used in statistics and machine learning for computationally efficient and flexible modeling of Gaussian random fields. The computational efficiency of the SPDE approach relies on finite element approximations combined with numerical quadrature and mass lumping, which enable sparse matrix methods during inference. Although many works have studied finite element approximations of fractional SPDEs, the effect of the mass lumping and quadrature approximations used in practice has not been fully analyzed. To fill this gap, we derive convergence rates for numerical approximations of fractional SPDEs based on finite element discretizations combined with numerical quadrature and mass lumping. Specifically, we obtain explicit convergence rates for the mean-squared error of the covariance function in a general framework that covers the main settings where mass lumping is used in the SPDE approach. We also analyze non-stationary variance-control factors of the form $L^β(τu)=\mathcal{W}$, where $τ$ is spatially varying, and derive covariance error estimates showing how the regularity of $τ$ affects the convergence rate. As specific examples, we provide results for random fields on bounded Euclidean domains, Riemannian manifolds, and metric graphs. Numerical experiments are presented that confirm the theoretical results.

math.NA↗

arXiv2025

Spectral Theory of Krein-Feller Type Operators and Applications in Stochastic Fractional Elliptic and Parabolic Equations

It has been shown that the space $C^{\infty}_{W,V}(\mathbb{T})$, introduced in Simas and Sousa (Potential Analysis, 2025), is the natural regularity space for solutions of the eigenvalue problem $Δ_{W,V} u = λu$ on the torus $\mathbb{T}$, where $Δ_{W,V} = \frac{d^{+}}{dV}\frac{d^{-}}{dW}$ is the Krein Feller operator in the case where $W$ and $V$ are strictly increasing and right continuous (respectively left continuous), possibly with dense sets of discontinuities. In this work we provide conditions ensuring that every function in $C^{\infty}_{W,V}(\mathbb{T})$, which may be highly discontinuous, admits a series expansion that generalizes the classical Taylor expansion. A central feature of our approach is that all proofs are nonstandard, since classical analytical and spectral arguments cannot be adapted to this singular setting. Using these methods we characterize the eigenvectors of $Δ_{W,V}$ in terms of generalized trigonometric functions and obtain an asymptotic lower bound for the associated eigenvalues. We also derive a sharp upper bound for the convergence exponent of these eigenvalues, and as a consequence we prove that $C^{\infty}_{W,V}(\mathbb{T})$ is a nuclear space. Further consequences include results on the asymptotic behavior of eigenvalues of compact operators and improvements in traceability. As a final application we establish existence results for generalized fractional stochastic and deterministic differential equations, as well as for parabolic stochastic partial differential equations acting on nuclear spaces.

math.AP↗