Cristian Inzunza
Español

Research

My work combines the analysis of variational formulations with their discretisation. These are the main lines, each with the articles that belong to it.

Fig. 1. Manufactured solution of Example 1 in Careaga, Gatica, Inzunza and Ruiz-Baier (IMA J. Numer. Anal., 2025): fluid pressure on the deformed configuration, with the undeformed domain dashed.

Poroelasticity and multiphysics coupling

I study Biot's poroelasticity model and its couplings with heat transfer, Poisson–Nernst–Planck electrochemical transport, and Brinkman and Forchheimer flow. The formulations use the total pressure and pseudostress-type unknowns, and are designed to remain stable in the nearly incompressible limit, where standard discretisations suffer from locking. Applications include deformable porous media in engineering and fluid flow in biological tissues.

Fig. 1.1. Ionic polarization in a charged porous medium for an applied voltage V = 0.9 (Example 3 of C. Inzunza, A pseudostress-based fully mixed finite element method for the stationary Biot–Poisson–Nernst–Planck system, submitted): electric potential, displacement magnitude and Darcy-flux magnitude. Recomputed for this page in primal form with the data of the example; the circle marks the charged region.
Fig. 1.2. Exact solution of Example 1 in J. Camaño, S. Caucao and C. Inzunza, A five-field mixed formulation for the fully dynamic Biot–Brinkman model (ESAIM: M2AN, accepted): shear stress, fluid-velocity and displacement magnitudes, and pore pressure at t = 0.

Related publications

  1. 2026

    A five-field mixed formulation for the fully dynamic Biot–Brinkman model

    J. Camaño, S. Caucao, C. Inzunza

    ESAIM: Mathematical Modelling and Numerical Analysis, accepted

    DOI 10.1051/m2an/2026094Preprint

  2. 2026

    A locking-free mixed-primal formulation for the Biot–Brinkman–Forchheimer model

    J. Camaño, S. Caucao, C. Inzunza

    Preprint 2026-9, GIANuC², Universidad Católica de la Santísima Concepción

    Preprint

  3. 2026

    A pseudostress-based fully mixed finite element method for the stationary Biot–Poisson–Nernst–Planck system

    C. Inzunza

    Submitted

    DOI 10.13140/RG.2.2.11969.31844

  4. 2026

    A pseudostress-based fully mixed formulation for the unsteady Biot–Brinkman model

    J. Camaño, S. Caucao, C. Inzunza

    Preprint, GIANuC², Universidad Católica de la Santísima Concepción

    Preprint

  5. 2025

    New Banach spaces-based mixed finite element methods for the coupled poroelasticity and heat equations

    J. Careaga, G. N. Gatica, C. Inzunza, R. Ruiz-Baier

    IMA Journal of Numerical Analysis, 45(4), 1936–1984

    DOI 10.1093/imanum/drae052Preprint

  6. 2025

    Primal-mixed finite element methods for the coupled Biot and Poisson–Nernst–Planck equations

    G. N. Gatica, C. Inzunza, R. Ruiz-Baier

    Computers & Mathematics with Applications, 186, 53–83

    DOI 10.1016/j.camwa.2025.03.004Preprint

Fig. 2. Velocity magnitude near the cavity in the channel of Example 4 in Caucao, Gatica, Gatica and Inzunza (J. Comput. Appl. Math., 2027), on the finest adapted mesh, recomputed for this page. The flow accelerates above the cavity and bends away from the low-porosity region.

Flow in porous media and incompressible fluids

Mixed finite element methods for Stokes, Navier–Stokes, Brinkman and Brinkman–Forchheimer equations, including convective terms, variable porosity and nearly incompressible elasticity. The analysis relies on Banach-space formulations in which the natural integrability of the nonlinear terms determines the functional setting.

Fig. 2.1. Layer 70 of the SPE10 benchmark, the heterogeneous medium of Example 2 in J. Camaño, S. Caucao and C. Inzunza, A pseudostress-based fully mixed formulation for the unsteady Biot–Brinkman model (GIANuC² preprint, 2026): permeability on the 60 × 220 grid, and the magnitude of a steady Darcy velocity for a unit pressure drop from left to right, computed here with a two-point flux scheme.

Related publications

  1. 2027

    A priori and a posteriori error analysis of a mixed FEM for stationary convective Brinkman–Forchheimer flows with variable porosity

    S. Caucao, G. N. Gatica, L. F. Gatica, C. Inzunza

    Journal of Computational and Applied Mathematics, 492, Article 118115

    Preprint

  2. 2026

    A five-field mixed formulation for the fully dynamic Biot–Brinkman model

    J. Camaño, S. Caucao, C. Inzunza

    ESAIM: Mathematical Modelling and Numerical Analysis, accepted

    DOI 10.1051/m2an/2026094Preprint

  3. 2026

    A locking-free mixed-primal formulation for the Biot–Brinkman–Forchheimer model

    J. Camaño, S. Caucao, C. Inzunza

    Preprint 2026-9, GIANuC², Universidad Católica de la Santísima Concepción

    Preprint

  4. 2026

    A pseudostress-based fully mixed formulation for the unsteady Biot–Brinkman model

    J. Camaño, S. Caucao, C. Inzunza

    Preprint, GIANuC², Universidad Católica de la Santísima Concepción

    Preprint

  5. 2023

    A Banach spaces-based fully-mixed finite element method for the stationary chemotaxis-Navier–Stokes problem

    S. Caucao, E. Colmenares, G. N. Gatica, C. Inzunza

    Computers & Mathematics with Applications, 145, 65–89

    DOI 10.1016/j.camwa.2023.06.006Preprint

  6. 2022

    A posteriori error analysis of Banach spaces-based fully-mixed finite element methods for Boussinesq-type models

    G. N. Gatica, C. Inzunza, R. Ruiz-Baier, F. Sandoval

    Journal of Numerical Mathematics, 30, 325–356

    DOI 10.1515/jnma-2021-0101Preprint

  7. 2021

    On the well-posedness of Banach spaces-based mixed formulations for the nearly incompressible Navier–Lamé and Stokes equations

    G. N. Gatica, C. Inzunza

    Computers & Mathematics with Applications, 102, 87–94

    DOI 10.1016/j.camwa.2021.10.004Preprint

  8. 2020

    An augmented fully-mixed finite element method for a coupled flow-transport problem

    G. N. Gatica, C. Inzunza

    Calcolo, 57

    DOI 10.1007/s10092-020-0355-yPreprint

Fig. 3. Initial cell density of Example 3 in Caucao, Colmenares, Gatica and Inzunza (Comput. Math. Appl., 2023): three clusters near the top of the domain, with level lines of the chemical signal.

Transport, diffusion and their coupling with mechanics

Coupled problems in which a concentration or temperature is transported by a flow or driven by mechanical stress: stress-assisted diffusion, chemotaxis–Navier–Stokes, Boussinesq-type models, flow–transport systems and heat-assisted sedimentation. Well-posedness is typically obtained through fixed-point arguments combined with the theory of mixed and monotone problems.

Fig. 3.1. Batch settling in the vessel of Example 2 in J. Careaga and C. Inzunza, Mixed finite element methods for a heat-assisted model of sedimentation (arXiv:2610.28903): solids volume fraction at four times from the initial value 0.3. Computed here for the solids equation alone, with the batch flux and diffusion functions of the paper and without the flow and heat coupling.

Related publications

  1. 2026

    Mixed finite element methods for a heat-assisted model of sedimentation

    J. Careaga, C. Inzunza

    arXiv preprint arXiv:2610.28903

    arXiv:2610.28903

  2. 2026

    A pseudostress-based fully mixed finite element method for the stationary Biot–Poisson–Nernst–Planck system

    C. Inzunza

    Submitted

    DOI 10.13140/RG.2.2.11969.31844

  3. 2025

    New Banach spaces-based mixed finite element methods for the coupled poroelasticity and heat equations

    J. Careaga, G. N. Gatica, C. Inzunza, R. Ruiz-Baier

    IMA Journal of Numerical Analysis, 45(4), 1936–1984

    DOI 10.1093/imanum/drae052Preprint

  4. 2025

    Primal-mixed finite element methods for the coupled Biot and Poisson–Nernst–Planck equations

    G. N. Gatica, C. Inzunza, R. Ruiz-Baier

    Computers & Mathematics with Applications, 186, 53–83

    DOI 10.1016/j.camwa.2025.03.004Preprint

  5. 2023

    New Banach spaces-based fully-mixed finite element methods for pseudostress-assisted diffusion problems

    G. N. Gatica, C. Inzunza, F. A. Sequeira

    Applied Numerical Mathematics, 193, 148–178

    DOI 10.1016/j.apnum.2023.07.017Preprint

  6. 2023

    A Banach spaces-based fully-mixed finite element method for the stationary chemotaxis-Navier–Stokes problem

    S. Caucao, E. Colmenares, G. N. Gatica, C. Inzunza

    Computers & Mathematics with Applications, 145, 65–89

    DOI 10.1016/j.camwa.2023.06.006Preprint

  7. 2022

    A posteriori error analysis of Banach spaces-based fully-mixed finite element methods for Boussinesq-type models

    G. N. Gatica, C. Inzunza, R. Ruiz-Baier, F. Sandoval

    Journal of Numerical Mathematics, 30, 325–356

    DOI 10.1515/jnma-2021-0101Preprint

  8. 2022

    A pseudostress-based mixed-primal finite element method for stress-assisted diffusion problems in Banach spaces

    G. N. Gatica, C. Inzunza, F. A. Sequeira

    Journal of Scientific Computing, 92

    DOI 10.1007/s10915-022-01959-9Preprint

  9. 2020

    An augmented fully-mixed finite element method for a coupled flow-transport problem

    G. N. Gatica, C. Inzunza

    Calcolo, 57

    DOI 10.1007/s10092-020-0355-yPreprint

Fig. 4. Porosity of the same channel on the finest mesh of the adaptive sequence. The refinement follows the cavity corners and the two low-porosity regions.

A posteriori error analysis and adaptivity

Residual-based a posteriori error estimators for fully-mixed and mixed-primal methods, with proofs of reliability and efficiency, and their use to drive adaptive mesh refinement in problems with singularities or sharp layers.

Fig. 4.1. Internal layer on a three-quarter star (Example 2 of A pseudostress-based fully mixed finite element method for the stationary Biot–Poisson–Nernst–Planck system): exact charge difference and electric-displacement magnitude on a mesh refined towards the layer along y = −x.

Related publications

  1. 2027

    A priori and a posteriori error analysis of a mixed FEM for stationary convective Brinkman–Forchheimer flows with variable porosity

    S. Caucao, G. N. Gatica, L. F. Gatica, C. Inzunza

    Journal of Computational and Applied Mathematics, 492, Article 118115

    Preprint

  2. 2026

    A pseudostress-based fully mixed finite element method for the stationary Biot–Poisson–Nernst–Planck system

    C. Inzunza

    Submitted

    DOI 10.13140/RG.2.2.11969.31844

  3. 2022

    A posteriori error analysis of Banach spaces-based fully-mixed finite element methods for Boussinesq-type models

    G. N. Gatica, C. Inzunza, R. Ruiz-Baier, F. Sandoval

    Journal of Numerical Mathematics, 30, 325–356

    DOI 10.1515/jnma-2021-0101Preprint