arXiv subjects

Patrick Vega

Publications and source records attributed to Patrick Vega.

5 recordsLinked to original sources

arXiv2026

An adaptive superconvergent mixed finite element method for exactly symmetric linear elasticity based on local residual minimization

We introduce an a posteriori error estimator for exactly symmetric mixed finite element discretizations of linear elasticity, based on recasting a Stenberg-type postprocessing scheme as a local residual minimization problem in a discrete dual norm. This construction yields, as a dual variable and at no additional computational cost, a Riesz representative of the associated local residual, from which we build an a posteriori error indicator. We establish a reliability estimate, with the dependence on the Lam\'e parameters tracked explicitly, and a local efficiency estimate, without auxiliary bubble functions, in the standard compressible regime, as well as an alternative reliability estimate, based on a robust stability property and an Oswald averaging operator, with a constant that remains bounded as $\lambda\to\infty$; the local efficiency estimate holds, with the same bounded behavior, uniformly in both regimes. Notably, a single indicator and a single comparison norm serve both regimes, in contrast with existing hypercircle-based estimators, which require a distinct construction for the incompressible limit. Numerical examples, including a benchmark with a known singular solution and one without an analytical solution, validate the theoretical findings.

math.NA↗

arXiv2025

A robust fully-mixed finite element method with skew-symmetry penalization for low-frequency poroelasticity

In this work, we present and analyze a fully-mixed finite element scheme for the dynamic poroelasticity problem in the low-frequency regime. We write the problem as a four-field, first-order, hyperbolic system of equations where the symmetry constraint on the stress field is imposed via penalization. This strategy is equivalent to adding a perturbation to the saddle point system arising when the stress symmetry is weakly-imposed. The coupling of solid and fluid phases is discretized by means of stable mixed elements in space and implicit time advancing schemes. The presented stability analysis is fully robust with respect to meaningful cases of degenerate model parameters. Numerical tests validate the convergence and robustness and assess the performances of the method for the simulation of wave propagation phenomena in porous materials.

math.NA↗

arXiv2023

Minimum-residual a posteriori error estimates for hybridizable discontinuous Galerkin discretizations of the Helmholtz equation

We propose and analyze two a posteriori error indicators for hybridizable discontinuous Galerkin (HDG) discretizations of the Helmholtz equation. These indicators are built to minimize the residual associated with a local superconvergent postprocessing scheme for the primal variable, measured in a dual norm of an enlarged discrete test space. The residual minimization is reformulated into equivalent local saddle-point problems, each yielding a superconvergent postprocessed approximation of the primal variable in the asymptotic regime for sufficiently regular exact solutions and a built-in residual representation with minimal computational effort. Both error indicators are based on frequency-dependent postprocessing schemes and verify reliability and efficiency estimates for a frequency-weighted $H^1$-error for the scalar variable and the $L^2$-error for the flux. We illustrate our theoretical findings through ad-hoc numerical experiments.

math.NA↗

arXiv2022

An adaptive superconvergent finite element method based on local residual minimization

We introduce an adaptive superconvergent finite element method for a class of mixed formulations to solve partial differential equations involving a diffusion term. It combines a superconvergent postprocessing technique for the primal variable with an adaptive finite element method via residual minimization. Such a residual minimization procedure is performed on a local postprocessing scheme, commonly used in the context of mixed finite element methods. Given the local nature of that approach, the underlying saddle point problems associated with residual minimizations can be solved with minimal computational effort. We propose and study a posteriori error estimators, including the built-in residual representative associated with residual minimization schemes; and an improved estimator which adds, on the one hand, a residual term quantifying the mismatch between discrete fluxes and, on the other hand, the interelement jumps of the postprocessed solution. We present numerical experiments in two dimensions using Brezzi-Douglas-Marini elements as input for our methodology. The experiments perfectly fit our key theoretical findings and suggest that our estimates are sharp.

math.NA↗

arXiv2015

A high order HDG method for curved-interface problems via approximations from straight triangulations

We generalize the technique of [Solving Dirichlet boundary-value problems on curved domains by extensions from subdomains, SIAM J. Sci. Comput. 34, pp. A497--A519 (2012)] to elliptic problems with mixed boundary conditions and elliptic interface problems involving a non-polygonal interface. We study first the treatment of the Neumann boundary data since it is crucial to understand the applicability of the technique to curved interfaces. We provide numerical results showing that, in order to obtain optimal high order convergence, it is desirable to construct the computational domain by interpolating the boundary/interface using piecewise linear segments. In this case the distance of the computational domain to the exact boundary is only $O(h^2)$.

math.NA↗