A new residual-type flux-free error estimator is presented. It computes upper and lower bounds of the error in energy norm with the ultimate goal of obtaining bounds for outputs of interest. The proposed approach precludes the main drawbacks of standard residual type estimators circumventing the need of flux-equilibration and resulting in a simple implementation that uses standard resources available in finite element codes. This is especially interesting for existing codes and 3D applications where the implementation of this technique is as simple as in 2D. Recall that on the contrary, the complexity of the flux-equilibration techniques increases drastically in the 3D case. Bounds for the energy norm of the error are used to produce upper and lower bounds of linear functional outputs, representing quantities of engineering interest. This new flux-free error estimator improves the effectivity of previous approaches (better accuracy in every test) and it can be used in the mechanical case for linear elements. The proposed approach demonstrates its efficiency in numerical tests producing sharp bounds of the reference error both for the energy and the quantities of interest.

Keywords: Separated representations, Finite sums decomposition, Error estimation, Curse of dimensionality High dimensional models Fine modeling of the structure and mechanics of materials from the nanometric to the micrometric scales uses descriptions ranging from quantum to statistical mechanics. Most of these models consist of a partial differential equation defined in a highly multidimensional domain (e.g. Schrodinger equation, FokkerPlanck equations among many others). The main challenge related to these models is their associated curse of dimensionality. We proposed in some of our former works a new strategy able to circumvent the curse of dimensionality based on the use of separated representations (also known as finite sums decomposition). This technique proceeds by computing at each iteration a new sum that consists of a product of functions each one defined in one of the model coordinates. The issue related to error estimation has never been addressed. This paper presents a first attempt on the accuracy evaluation of such a kind of discretization techniques.

Abstract. The paper introduces a methodology to compute strict upper and lower bounds for linear-functional outputs of the exact solutions of the advection-reaction-diffusion equation. The proposed approach is an alternative to the standard residual type estimators (hybrid-flux), circumventing the need of flux-equilibration following a fluxfree error estimation strategy. The presented estimator provides sharper estimates than the ones provided by both the standard hybrid-flux techniques and other flux-free techniques.

Colloidal crystals of poly(styrene) spheres can be used as templates to fabricate a conjugated polymer inverse opal potentiometric biosensor (see Figure) incorporating an enzyme. After core removal, a seven‐fold increase in sensitivity relative to conventional films is measured for analyte concentrations varying from 1 to 100 μM. The presence of large pores enables the formation of a Schottky junction, which governs the sensing response.

SUMMARYThe solution of a steady thermal multiphase problem is assumed to be dependent on a set of parameters describing the geometry of the domain, the internal interfaces and the material properties. These parameters are considered as new independent variables. The problem is therefore stated in a multidimensional setup. The Proper Generalized Decomposition (PGD) provides an approximation scheme especially well suited to preclude dramatically increasing the computational complexity with the number of dimensions. The PGD strategy is reviewed for the standard case dealing only with material parameters. Then, the ideas presented in [Ammar et al., "Parametric solutions involving geometry: A step towards efficient shape optimization." Comput. Methods Appl. Mech. Eng., 2014; 268:178-193] to deal with parameters describing the domain geometry are adapted to a more general case including parametrization of the location of internal interfaces. Finally, the formulation is extended to combine the two types of parameters. The proposed strategy is used to solve a problem in applied geophysics studying the temperature field in a cross section of the Earth crust subsurface. The resulting problem is in a 10-dimensional space, but the PGD solution provides a fairly accurate approximation (error  1%) using less that 150 terms in the PGD expansion.