Equilibrated tractions for the Hybrid High-Order method

We show how to recover equilibrated face tractions for the hybrid high-order method for linear elasticity recently introduced in [D. A. Di Pietro and A. Ern, A hybrid high-order locking-free method for linear elasticity on general meshes, Comput. Meth. Appl. Mech. Engrg., 2015, 283:1-21], and prove that these tractions are optimally convergent

Error analysis of discontinuous Galerkin discretizations of a class of linear wave-type problems

In this paper we consider central fluxes discontinuous Galerkin space discretizations of a general class of wave-type equations of Friedrichs’ type. This class includes important examples such as Maxwell’s equations and wave equations. We prove an optimal error bound which holds under suitable regularity assumptions on the solution. Our analysis is performed in a framework of evolution equations on a Hilbert space and thus allows for the combination with various time integration schemes

A posteriori error control for discontinuous Galerkin methods for parabolic problems

We derive energy-norm a posteriori error bounds for an Euler time-stepping method combined with various spatial discontinuous Galerkin schemes for linear parabolic problems. For accessibility, we address first the spatially semidiscrete case, and then move to the fully discrete scheme by introducing the implicit Euler time-stepping. All results are presented in an abstract setting and then illustrated with particular applications. This enables the error bounds to hold for a variety of discontinuous Galerkin methods, provided that energy-norm a posteriori error bounds for the corresponding elliptic problem are available. To illustrate the method, we apply it to the interior penalty discontinuous Galerkin method, which requires the derivation of novel a posteriori error bounds. For the analysis of the time-dependent problems we use the elliptic reconstruction technique and we deal with the nonconforming part of the error by deriving appropriate computable a posteriori bounds for it.Comment: 6 figure

Asymptotic optimality of the edge finite element approximation of the time-harmonic Maxwell's equations

We analyze the conforming approximation of the time-harmonic Maxwell's equations using N\'ed\'elec (edge) finite elements. We prove that the approximation is asymptotically optimal, i.e., the approximation error in the energy norm is bounded by the best-approximation error times a constant that tends to one as the mesh is refined and/or the polynomial degree is increased. Moreover, under the same conditions on the mesh and/or the polynomial degree, we establish discrete inf-sup stability with a constant that corresponds to the continuous constant up to a factor of two at most. Our proofs apply under minimal regularity assumptions on the exact solution, so that general domains, material coefficients, and right-hand sides are allowed

An unfitted hybrid high-order method for the stokes interface problem

We design and analyze a hybrid high-order method on unfitted meshes to approximate the Stokes interface problem. The interface can cut through the mesh cells in a very general fashion. A cell-agglomeration procedure prevents the appearance of small cut cells. Our main results are inf-sup stability and a priori error estimates with optimal convergence rates in the energy norm. Numerical simulations corroborate these results

A priori and a posteriori analysis of non-conforming finite elements with face penalty for advection-diffusion equations

We analyse a non-conforming finite-element method to approximate advection-diffusion-reaction equations. The method is stabilized by penalizing the jumps of the solution and those of its advective derivative across mesh interfaces. The a priori error analysis leads to (quasi-)optimal estimates in the mesh size (sub-optimal by order ½ in the L2-norm and optimal in the broken graph norm for quasi-uniform meshes) keeping the Péclet number fixed. Then, we investigate a residual a posteriori error estimator for the method. The estimator is semi-robust in the sense that it yields lower and upper bounds of the error which differ by a factor equal at most to the square root of the Péclet number. Finally, to illustrate the theory we present numerical results including adaptively generated meshe

Applications of nonvariational finite element methods to Monge--Amp\`ere type equations

The goal of this work is to illustrate the application of the nonvariational finite element method to a specific Monge--Amp\`ere type nonlinear partial differential equation. The equation we consider is that of prescribed Gauss curvature.Comment: 7 pages, 3 figures, tech repor

Sequential Injection-flow Reversal Mixing (Si-frm) Untuk Penentuan Kreatinin Dalam Urin

Jumlah kreatinin yang diekskresikan melalui urin menunjukkan keadaan ginjal seseorang. Dalam penelitian ini, dikembangkan metode untuk penentuan kreatinin secara otomatis yaitu sequential injection-flow reversal mixing (SI-FRM). Pendeteksian kreatinin didasarkan pada pembentukan senyawa berwarna (merah-orange) yang dihasilkan dari reaksi antara kreatinin dan asam pikrat dalam suasana basa dan diukur pada panjang gelombang 530 nm. Reaksi pembentukan senyawa kreatinin-pikrat dilakukan melalui pembentukan segmen antara sampel dan reagen di-holding coil dan selanjutnya dilakukan proses flow reversal di-mixing coil. Parameter-parameter yang mempengaruhi metode ini diuji secara detail. Hasil penelitian menunjukkan bahwa kondisi optimum pengukuran kreatinin yaitu menggunakan konsentrasi asam pikrat 0,035 M dan NaOH 3,5%, laju alir flow reversal 5 µL/detik, laju alir produk reaksi 20 µL/detik, jumlah flow reversal empat kali dan menggunakan tiga segmen (pikrat-kreatinin-pikrat) dengan masing-masing volume segmen 100 µL. Metode SI-FRM ini telah diaplikasikan langsung untuk penentuan kadar kreatinin dalam urin dengan limit deteksi 1,7 µg/g. The amount of creatinine excreted in urine indicates kidney condition. In this experiment, the automatic determination method of determining creatinine was developed by using sequential injection-flow reversal mixing (SI-FRM). The detection of creatinine is based on the formation of a colored product (red-orange) yielded from the reaction of creatinine with picrate at alkaline medium. The absorbance is measured at wavelength of 530 nm.  The formation of creatinine-picrate complex is performed through the segment formation between sample and reagent in the holding coil and then flow reversal process in the mixing coil of SI-FRM. Several parameters affecting to this method are investigated in detail. The results show that the optimum concentrations of picric acid and NaOH are 0.035 M and 3.5%, respectively. Other optimized conditions, such as the flow reversal rate of there 5 µL/s, flow rate of product of 20 µL/s, amount of flow reversal process of four times, and segment amount of three (picrate-creatinine -picrate) with each volume of 100 µL, were obtained. This method is successfully applied to the determination of creatinine in urine with the detection limit of 1.7 µg/g

A scalable parallel finite element framework for growing geometries. Application to metal additive manufacturing

This work introduces an innovative parallel, fully-distributed finite element framework for growing geometries and its application to metal additive manufacturing. It is well-known that virtual part design and qualification in additive manufacturing requires highly-accurate multiscale and multiphysics analyses. Only high performance computing tools are able to handle such complexity in time frames compatible with time-to-market. However, efficiency, without loss of accuracy, has rarely held the centre stage in the numerical community. Here, in contrast, the framework is designed to adequately exploit the resources of high-end distributed-memory machines. It is grounded on three building blocks: (1) Hierarchical adaptive mesh refinement with octree-based meshes; (2) a parallel strategy to model the growth of the geometry; (3) state-of-the-art parallel iterative linear solvers. Computational experiments consider the heat transfer analysis at the part scale of the printing process by powder-bed technologies. After verification against a 3D benchmark, a strong-scaling analysis assesses performance and identifies major sources of parallel overhead. A third numerical example examines the efficiency and robustness of (2) in a curved 3D shape. Unprecedented parallelism and scalability were achieved in this work. Hence, this framework contributes to take on higher complexity and/or accuracy, not only of part-scale simulations of metal or polymer additive manufacturing, but also in welding, sedimentation, atherosclerosis, or any other physical problem where the physical domain of interest grows in time