This volume comprises the first part of the proceedings of the 10th International Conference on Finite Volumes for Complex Applications, FVCA, held in Strasbourg, France, during October 30 to November 3, 2023. The Finite Volume method, and several of its variants, is a spatial discretization technique for partial differential equations based on the fundamental physical principle of conservation. Recent decades have brought significant success in the theoretical understanding of the method. Many finite volume methods are also built to preserve some properties of the continuous equations, including maximum principles, dissipativity, monotone decay of the free energy, asymptotic stability, or stationary solutions. Due to these properties, finite volume methods belong to the wider class of compatible discretization methods, which preserve qualitative properties of continuous problems at the discrete level. This structural approach to the discretization of partial differential equations becomes particularly important for multiphysics and multiscale applications. In recent years, the efficient implementation of these methods in numerical software packages, more specifically to be used in supercomputers, has drawn some attention.   This volume contains all invited papers, as well as the contributed papers focusing on finite volume schemes for elliptic and parabolic problems. They include structure-preserving schemes, convergence proofs, and error estimates for problems governed by elliptic and parabolic partial differential equations. The second volume is focused on finite volume methods for hyperbolic and related problems, such as methods compatible with the low Mach number limit or able to exactly preserve steady solutions, the development and analysis of high order methods, or the discretization of kinetic equations.

Table of Contents:
Invited papers, R. Abgrall, A personal discussion on conservation, and how to formulate it.- W. Boscheri, C. Birke and C. Klingenberg, A high order semi-implicit scheme for ideal magnetohydrodynamics.- A. Artoni, P. F. Antonietti, R. Corradi, I. Mazzieri, N. Parolini, D. Rocchi, P. Schito and Francesco F. Semeraro, AeroSPEED: a high order acoustic solver for aeroacoustic applications.- C. Cancès, M. Herda and A. Massimini, Finite volumes for a generalized Poisson-Nernst-Planck system with crossdiffusion and size exclusion.- X. D. Sánchez and J. Ryan, Magic SIAC Toolbox: A Codebase of Effective, Efficient, and Flexible Filters.- C. Helzel and E. Chudzik, A Review of Cartesian Grid Active Flux Methods for Hyperbolic Conservation Laws.- C. Rohde, Moving-Mesh Finite-Volume Methods for Hyperbolic Interface Dynamics.- M. Peszynska, Mixed dimensional modeling with overlapping continua on Cartesian grids for complex applications.- Contributed papers: Pierre-Loïc Bacq, Antoine Gerschenfeldand Michael Ndjinga, PolyMAC: staggered finite volume methods on general meshes for incompressible Navier-Stokes problems.- C. Bauzet, F. Nabet, K. Schmitz and A. Zimmermann, Finite Volume Approximations for Non-Linear Parabolic Problems with Stochastic Forcing.- F. Benkhaldoun and Abdallah Bradji, A new analysis for a super-convergence result in the divergence norm for Lowest Order Raviart-Thomas Mixed Finite Elements combined with the Crank-Nicolson method applied to one dimensional parabolic equations.- Benkhaldoun, Fayssal, Bradji, Abdallah, An L∞(H1)–error estimate for Gradient Schemes applied to time fractional diffusion equations.- Jerome Bonelle and Thomas Fonty, Compatible Discrete Operator schemes for solidification and segregation phenomena.- M. Boutilier, K. Brenner and V. Dolean, Trefftz approximation space for Poisson equation in perforated domains.- C. Cancès, J. Cauvin-Vila, C. Chainais-Hillairet and V. Ehrlacher, Structure Preserving Finite Volume Approximation of Cross-Diffusion Systems Coupled by a Free Interface.- C. Chainais-Hillairet and M. Alfaro, Finite volume scheme for the diffusive field-road model: study of the long time behaviour.- C. Chainais-Hillairet, R. Eymard and J. Fuhrmann, An approximate two-point Dirichlet flux for quasilinear convection diffusion equations.- Z. Chehade and Y. Coudière, The Two-Point Finite Volume Scheme for the Microscopic Bidomain Model of Electrocardiology.- E. Chénier, C. Le Potier, Erell Jamelot and Andrew Peitavy, Improved Crouzeix-Raviart scheme for the Stokes problem.- S. Clément, F. Lemarié and E. Blayo, Towards a finite volume discretization of the atmospheric surface layer consistent with physical theory.- J. Droniou, M. Laaziri and R. Masson, Thermodynamically Consistent discretisation of a Thermo-HydroMechanical model.- E. Eggenweiler, J. Nickl and I. Rybak, Justification of Generalized Interface Conditions for Stokes-Darcy Problems.- J. Fuhrmann, B. Gaudeul and C. Keller, Two entropic finite volume schemes for a Nernst–Planck–Poisson system with ion volume constraints.- M. Gander, J. Hennicker, R. Masson and T. Vanzan, Dimensional reduction by Fourier analysis of a Stokes-Darcy fracture model.- M. Heida, Finite Volumes for Simulation of Large Molecules.- M. M. Knodel, Arne Nägel, Eva Herrmann and Gabriel Wittum, PDE models of virus replication merging 2D manifold and 3D volume effects evaluated at realistic reconstructed cell geometries.- S. Krell and J. Moatti, Structure-preserving schemes for drift-diffusion systems on general meshes: DDFV vs HFV.- S. Matera, D. Runge and C. Merdon, Reduced Basis Approach for convection-diffusion equations with non-linear boundary reaction conditions.- J. Moatti, A skeletal high-order structure preserving scheme for advection-diffusion equations.- G. Narváez, M. Ferrand, T. Fonty and S. Benhamadouche, Automatic solid reconstructionfrom 3-D points set for flow simulation via an immersed boundary method.- L. Ruan and I. Rybak, Stokes–Brinkman–Darcy Models for Coupled Free-Flow and Porous-Medium Systems.- P. Strohbeck, C. Riethmüller, D. Göddeke and I. Rybak, Robust and Efficient Preconditioners for Stokes–Darcy Problems.- C. Thomas, S. Mazen and El-Houssaine Quenjel, A DDFV Scheme for Incompressible Two-Phase Flow Degenerate Problem in Porous Media.- Author Index.