We propose a new family of multiscale hybrid mixed methods (MHM) for the reactiveadvective-diffusive (RAD) equation in complex domains. It generalizes the MHM methods originally proposed in Harder, Paredes and Valentin (2013 and 2015) to polytopal meshes and covers all asymptotic regimes of the model within a single mathematical framework. As a result, the skeletal MHM method changes its structure automatically, from primal to mixed forms, depending on the asymptotic of local RAD solutions, which respond to multiscale basis functions at the element level. We establish the existence, uniqueness, and optimality of the MHM solution with respect to two -scale mesh parameters, relating it to the solution of a discrete primal hybrid version of the RAD model. Furthermore, we estimate the condition number of the matrices associated with the local problems responsible for upscaling, from which we establish upper limits for the condition number of the algebraic system associated with the MHM method. Numerical experiments validate theoretical results.

For a reaction-dominated diffusion problem we study a primal and a dual hybrid finite element method where weak continuity conditions are enforced by Lagrange multipliers. Uniform robustness of the discrete methods is achieved by enriching the local discretization spaces with modified face bubble functions which decay exponentially in the interior of an element depending on the ratio of the singular perturbation parameter and the local mesh-size. A posteriori error estimators are derived using Fortin operators. They are robust with respect to the singular perturbation parameter. Numerical experiments are presented that show that oscillations, if present, are significantly smaller then those observed in common finite element methods.

In this work we propose, analyze, and test a new multiscale finite element method called Multiscale Hybrid (MH) method. The method is built as a close relative to the Multiscale Hybrid Mixed (MHM) method, but with the fundamental difference that a novel definition of the Lagrange multiplier is introduced. The practical implication of this is that both the local problems to compute the basis functions, as well as the global problem, are elliptic, as opposed to the MHM method (and also other previous methods) where a mixed global problem is solved and constrained local problems are solved to compute the local basis functions. The error analysis of the method is based on a hybrid formulation, and a static condensation process is done at the discrete level, so the final global system only involves the Lagrange multipliers. We tested the performance of the method by means of numerical experiments for problems with multiscale coefficients, and we carried out comparisons with the MHM method in terms of performance, accuracy, and memory requirements.

This work proposes a new finite element for the multiscale hybrid-mixed method (MHM) applied to the Poisson equation with highly oscillatory coefficients. Unlike the original MHM method, multiscale bases are the solution to local Neumann problems driven by piecewise continuous polynomial interpolation on the skeleton faces of the macroscale mesh. As a result, we prove the optimal convergence of MHM by refining the face partition and leaving the mesh of macroelements fixed. This property allows the MHM method to be resonance free under the usual assumptions of local regularity. The numerical analysis of the method also revisits and complements the original approach proposed by D. Paredes, F. Valentin and H. Versieux (2017). Numerical experiments assess the new theoretical results.

The objective of this study was to identify transcripts or hormone-based biomarkers to define the physiological age of ' Hass ' avocado fruit and to elucidate the changes at the level of metabolic pathways and their regulation. ' Hass ' avocado fruit from orchards in different agroclimatic zones were collected during two harvest periods. Fruit were stored for 30 d under controlled atmosphere and regular air conditions and then transferred to shelflife conditions at 20 degrees C. The physiological age as represented by the initial state of a hypothetical enzyme system (E0) of each fruit was obtained through a mechanistic softening model for Chilean ' Hass ' avocado previously developed. Fruit from three different E0 ranges (low, intermediate and high) were selected for transcriptome and hormone analyses. Sequencing data were processed by partial least squares regression analysis, which revealed 46 genes correlated to E0. Different metabolic pathways were over expressed between low and high E0 fruit. Low E0 fruit showed overexpression of genes related to DNA replication, auxin transport, cell wall remodeling, gibberellin synthesis, brassinosteroids and flavonols. On the other hand, fruit with high E0 revealed genes related to ethylene and abscisic acid biosynthesis and related responses and phenylpropanoid biosynthesis. Likewise, targeted hormone analysis revealed higher concentrations of active gibberellins (GA1 and GA4) and jasmonic acid for low E0 fruit and for high E0 fruit higher concentrations of abscisic acid, salicylic acid, dihydrozeatin, indole acetic acid and trans-zeatin, only the latter two being significant for this phenotype. This study reveals the relationship between transcripts and hormones during fruit maturation that is key to evaluate the physiological age of ' Hass ' avocado fruit.

The aim of this study was to model Chilean "Hass" avocado softening behaviour, destined to local and distant markets, taking into account the biological variation given by growing location and harvest stages. A total of 24 batches were obtained during the season 2018-2019 from different agro-climatic zones (coast, intermediate and interior) and two harvest stages (based on dry matter content). Fruit softening during either regular air (RA) or controlled atmosphere (CA) storage at 5 degrees C followed by shelf-life at 20 degrees C was modelled using a simplified mechanistic model. Most of the model parameters were treated as being generic for all fruit except for two fruit specific parameters, F-0 (firmness at harvest) and E-0 (amount of enzyme complex at harvest) that characterized the fruit at harvest and thus postharvest ripening behaviour. The model was able to describe 87.6 % of the observed variation of all 24 fruit batches studied from different agro-climatic zones at the batch averaged level, but 93.5 % of the observed variation at the fruit individual level. Since measured at harvest when most fruit are highly firm, initial fruit firmness by itself was not able to discriminate among the various batches as they all showed similar normal distributions among the different agro-climatic zones, in addition, the estimated E-0 values for each individual fruit were correlated to key metabolites to identify potential metabolite biomarkers discriminating among the different regions and batches. The developed model can be utilized to predict the batch specific ripening behaviour of "Hass" avocado under different postharvest logistic chains given the distribution of E-0 is known.

This work extends the general form of the multiscale hybrid-mixed (MHM) method for the second-order Laplace (Darcy) equation to general non-conforming polygonal meshes. The main properties of the MHM method, i.e., stability, optimal convergence, and local conservation, are proven independently of the geometry of the elements used for the first level mesh. More precisely, it is proven that piecewise polynomials of degree k and $$k+1$$, $$k \ge 0$$, for the Lagrange multipliers (flux), along with continuous piecewise polynomial interpolations of degree $$k+1$$ posed on second-level sub-meshes are stable if the latter is fine enough with respect to the mesh for the Lagrange multiplier. We provide an explicit sufficient condition for this restriction. Also, we prove that the error converges with order $$k+1$$ and $$k+2$$ in the broken $$H^1$$ and $$L^2$$ norms, respectively, under usual regularity assumptions, and that such estimates also hold for non-convex; or even non-simply connected elements. Numerical results confirm the theoretical findings and illustrate the gain that the use of multiscale functions provides.

In this talk the recent extension of the Multiscale Hybrid-Mixed (MHM) method, originally proposed in [1], to the case of general polygonal meshes (that can be non-convex and non-conforming as well) will be presented. We present new stable multiscale finite elements such that they preserve the well-posedness, super-convergence and local conservation properties of the original MHM method under mild regularity conditions on the polygons. More precisely, we show that piecewise polynomial of degree k−1 and k, k≥1, for the Lagrange multipliers (flux) along with continuous piecewise polynomial interpolations of degree k posed on second-level sub-meshes are stable if the latter is refined enough. Such one- and two-level discretization impact the error in a way that the discrete primal (pressure) and dual (velocity) variables achieve super-convergence in the natural norms under extra local regularity only. Numerical tests illustrate theoretical results and the flexibility of the approach.

In this work, we address time dependent wave propagation problems with strong multiscale features (in space and time). Our goal is to design a family of innovative high performance numerical methods suitable to the simulation of such multiscale problems. Particularly, we extend the multiscale hybrid-mixed (MHM) finite element method for the two- and three-dimensional time-dependent Maxwell equations with heterogeneous coefficients. The MHM method arises from the decomposition of the exact electric and magnetic fields in terms of the solutions of locally independent Maxwell problems tied together with a one-field formulation on top of a coarse-mesh skeleton. The multiscale basis functions, which are responsible for upscaling, are driven by local Maxwell problems with tangential component of the magnetic field prescribed on faces. A high-order discontinuous Galerkin method in space combined with a second-order explicit leap-frog scheme in time discretizes the local problems. This makes the MHM method effective and yields a staggered algorithm within a “divide-and-conquer” framework. Several two-dimensional numerical tests assess the optimal convergence of the MHM method and its capacity to preserve the energy principle, as well as its accuracy to solve heterogeneous media problems on coarse meshes.

Antônio Tadeu Azevedo Gomes Weslley da Silva Pereira Roberto Pinto Souto Frédéric Valentin atagomes@lncc.br weslleyp@lncc.br rpsouto@lncc.br valentin@lncc.br Laboratório Nacional de Computação Cientı́fica (LNCC) Av. Getúlio Vargas, 333 Quitandinha, 25651-075, Petrópolis, RJ, Brazil Diego Paredes Concha diego.paredes@pucv.cl Pontificia Universidad Católica de Valparaı́so Av. Brasil 2950, 2340025 Valparaı́so, Región de Valparaı́so, Chile Abstract. We present a family of multiscale finite element methods for the linear elastodynamic model with highly heterogeneous coefficients, named Multiscale Hybrid-Mixed (MHM) methods. The MHM method consists of a strategy that naturally incorporates multiple scales in the numerical solutions while providing solutions with high-order precision for the primal and dual variables. It is a consequence of a hybridization procedure, which characterizes the unknowns as a direct sum of a coarse solution and the solutions to local problems with boundary conditions driven by the Lagrange multipliers. The completely independent local problems are embedded in the upscaling procedure, and computational approximations may be naturally obtained in a parallel computational environment. Numerical results verify the optimal convergence of the method, and its capacity to accurately incorporate heterogeneity and high-contrast coefficients in the numerical solution.

Antônio Tadeu Azevedo Gomes Weslley da Silva Pereira Roberto Pinto Souto Frédéric Valentin atagomes@lncc.br weslleyp@lncc.br rpsouto@lncc.br valentin@lncc.br LNCC Laboratório Nacional de Computação Cientı́fica 25651-075 Petrópolis, RJ, Brasil Diego Paredes diego.paredes@pucv.cl PUCV Pontificia Universidad Catolica de Valparaiso Valparaiso, Chile Abstract. We present the performance analysis of a parallel simulator that implements the multiscale hybrid-mixed (MHM) finite element method for the elastostatic equation. The formulation behind MHM comprises a global problem defined over the skeleton of a coarse mesh and a set of independent elasticity problems posed over element-wise submeshes. The performance analysis considers an isotropic elastic 3D model. For the strong scaling, we characterize whether it is more efficient to refine the submeshes and use polynomial degrees of lower order or vice-versa while keeping the same approximation error. We obtained the better efficiency— greater than 70% on 3,072 cores— when using higher-order polynomials. For the weak scaling, we fixed the high-order polynomials and varied the level of the refinement of the submeshes. We then increased the size of the problem by refining the global mesh at the same rate as the increase in the number of cores. The measurements suggest the configurations with refined submeshes offer the best efficiency— greater than 80% on 3,072 cores.

The family of Multiscale Hybrid-Mixed (MHM) finite element methods has received considerable attention from the mathematics and engineering community in the last few years. The MHM methods allow solving highly heterogeneous problems on coarse meshes while providing solutions with high-order precision. It embeds independent local problems which are responsible for upscaling unresolved scales into the numerical solution. These local contributions are brought together through a global problem defined on the skeleton of the coarse partition. Since the local problems are completely independent, they can be easily computed in parallel. In this paper, we present two simulator prototypes specifically crafted for the MHM methods, which adopt two different implementation strategies: (i) a multi-programming language approach, each language tackling different simulation issues; and (ii) a classical, single-programming language approach. Specifically, we use C++ for numerical computation of the global and local problems in a modular way; for process distribution in the simulator, we adopt the Erlang concurrent language in the first approach, and the MPI standard in the second approach. The aim of exploring these different approaches is twofold: (i) allow for the deployment of the simulator both in high-performance computing (with MPI) and in cloud computing environments (with Erlang); and (ii) pave the way for further exploration of quality attributes related to software productivity and fault-tolerance, which are key to Exascale systems. We present a performance evaluation of the two simulator prototypes taking into account their efficiency.

This work proposes a Multiscale Hybrid-Mixed (MHM) method for the Maxwell equation in time domain. The MHM method is a consequence of a hybridization procedure, and emerges as a method that naturally incorporates multiple scales while provides solutions with high-order precision. The computation of local problems is embedded in the upscaling procedure, which are completely independent and thus may be naturally obtained using parallel computation facilities. In this talk, we present the new MHM method for the two-dimensional Maxwell equations in time domain (Transverse Magnetic mode). We address some theoretical aspects of the method and propose an extensive numerical validation. We conclude that the MHM method is naturally shaped to be used in parallel computing environments and appears to be a highly competitive option to handle realistic multiscale hyperbolic boundary value problems with precision on coarse meshes.

In this work, we are interested in the propagation of electromagnetic waves in complex media. More precisely, we would like to study time dependent wave propagation problems with strong multiscale features (possibly in space and time). In this context we would like to contribute in the design of innovative numerical methods particularly well suited to the simulation of such problems. Indeed when a PDE model is approximated via classical finite element type method, it may suffer from a loss of accuracy when the solution presents multiscale features on coarse meshes. To address this issue, we rely on the concept of multiscale basis functions that is one solution to allow for accuracy even on coarse meshes. These basis functions are defined via algebraic relations. Contrary to classical polynomial approximation, they render by themselves a part of the high-contrast features of the problem at hand. Recently, a new family of finite element methods has been introduced in [1]-[2], referred as Multiscale Hybrid-Mixed methods (MHM), which is well adapted to the simulation of high-contrast or heterogeneous problems. The underlying approach relies on a two level discretization. Shortly, basis functions computed on a fine (second level) mesh allow for the reconstruction of the solution on a coarse (first level) mesh. Such MHM have been initially designed in the context of stationary problems, such as Darcy flows. In this work, we propose to extend the concept of MHM to time dependent electromagnetic wave propagation problems. The model problem relies on the time dependent Maxwell's equations. The continuity of the electric field is relaxed via the introduction of a Lagrange multiplier. The solutions are expressed on a basis computed at the second level that incorporates the heterogeneity of the problem via the resolution of a PDE. Several schemes are proposed from implicit to explicit time schemes and continuous finite elements to discontinuous ones for the spatial discretization of the local problems at the second level.

A new family of finite element methods, named Multiscale Hybrid-Mixed method (or MHM for short), aims to solve reactive-advective dominated problems with multiscale coefficients on coarse meshes. The underlying upscaling procedure transfers to the basis functions the responsibility of achieving high orders of accuracy. The upscaling is built inside the general framework of hybridization, in which the continuity of the solution is relaxed a priori and imposed weakly through the action of Lagrange multipliers. This characterizes the unknowns as the solutions of local problems with Robin boundary conditions driven by the multipliers. Such local problems are independent of one another, yielding a process naturally shaped for parallelization and adaptivity. Moreover, the multiscale decomposition indicates a new adaptive algorithm to set up local spaces defined using a face-based a posteriori error estimator. Interestingly, it also embeds a postprocessing of the dual variable (flux) which preserves local conservation properties of the exact solution. Extensive numerical validations assess the claimed optimal rates of convergence, the robustness of the method with respect to the model's coefficients, and the adaptivity algorithm.

In this work, we are interested in the propagation of electromagnetic waves in complex media. More precisely, we would like to study time dependent wave propagation problems with strong multiscales features (possibly in space and time). In this context we would like to contribute in the design of innovative numerical methods particularly well-suited to the simulation of such problems. Indeed when a PDE model is approximated via classical finite element type method, it may suffer from a loss of accuracy when the solution presents multiscales features on coarse meshes. To address this problematic, we rely on the concept of multiscale basis functions that is one solution to allow for accuracy even on coarse meshes. These basis functions are defined via algebraic relations. Contrary to classical polynomial approximation, they render by themselves a part of the high-contrast features of the problem at hand. Recently, researchers at LNCC (Laboratorio Nacional de Computacao Cientifica, Brasil) have introduced a family of finite element methods [1]-[2], called Multiscale Hybrid-Mixed methods (MHM), which is particularly well adapted to be used in high-contrast or heterogeneous problems.. The algorithm rely on a two level discretization. The basis function being computed at the second (finer) level allow for the reconstruction of the solution via the communication at the first (coarser) level on the squeleton of the mesh. This type of family has been firstly designed in the context of stationary problems, such as Darcy equations. In this work, we propose to extend the use of this Finite Element family to time dependent wave propagation problems. The model problem relies on the time dependent Maxwell's equations. The continuity of the electric field is relaxed via the introduction of a Lagrange multiplier. The solutions are expressed on a basis computed at the second level that incorporates the heterogeneity of the problem via the resolution of a PDE. Several schemes are proposed from implicit to explicit time schemes and continuous finite elements to discontinuous ones for the space resolution of the local problem at the second level. We propose some results on the validity of the algorithm from both theoretical and numerical point of view and will present first numerical results in 2D.

We aim at proposing novel stable finite element methods for the mixed Darcy equation with heterogeneous coefficients within a space splitting framework. We start from the primal hybrid formulation of the elliptic model for the pressure. Localization of this infinite-dimensional problem leads to element-level boundary value problems which embed multiscale and high-contrast features in a natural way, with Neumann boundary conditions driven by the Lagrange multipliers. Such a procedure leads to methods involving the space of piecewise constants for the pressure together with a discretization of the fluxes. Choosing (arbitrarily) polynomial interpolations, the lowest-order Raviart–Thomas element as well as some recent multiscale methods are recovered. In addition, the methods assure local mass conservation and can be interpreted as stabilized primal hybrid methods. Extensive numerical validation attests to the accuracy of the new methods on academic and more realistic problems with rough coefficients.

This work presents a priori and a posteriori error analyses of a new multiscale hybrid-mixed method (MHM) for anelliptic model. Specially designed to incorporate multiple scales into the construction of basis functions, thisfinite element method relaxes the continuity of the primal variable through the action of Lagrange multipliers, whileassuring the strong continuity of the normal component of the flux (dual variable). As a result, the dual variable,which stems from a simple postprocessing of the primal variable, preserves local conservation. We prove existence anduniqueness of a solution for the MHM method as well as optimal convergence estimates of any order in the naturalnorms. Also, we propose a face-residual a posteriori error estimator, and prove that it controls the error of bothvariables in the natural norms. Several numerical tests assess the theoretical results.