In this paper, we propose a multiphysics finite element method for a quasi-static thermo-poroelasticity model with a nonlinear convective transport term. To design some stable numerical methods and reveal the multi-physical processes of deformation, diffusion and heat, we introduce three new variables to reformulate the original model into a fluid coupled problem. Then, we introduce an Newton's iterative algorithm by replacing the convective transport term with $\nabla T^{i}\cdot(\bm{K}\nabla p^{i-1})$, $\nabla T^{i-1}\cdot(\bm{K}\nabla p^{i})$ and $\nabla T^{i-1}\cdot(\bm{K}\nabla p^{i-1})$, and apply the Banach fixed point theorem to prove the convergence of the proposed method. Then, we propose a multiphysics finite element method with Newton's iterative algorithm, which is equivalent to a stabilized method, can effectively overcome the numerical oscillation caused by the nonlinear thermal convection term. Also, we prove that the fully discrete multiphysics finite element method has an optimal convergence order. Finally, we draw conclusions to summarize the main results of this paper.
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We consider the task of estimating functions belonging to a specific class of nonsmooth functions, namely so-called tame functions. These functions appear in a wide range of applications: training deep learning, value functions of mixed-integer programs, or wave functions of small molecules. We show that tame functions are approximable by piecewise polynomials on any full-dimensional cube. We then present the first ever mixed-integer programming formulation of piecewise polynomial regression. Together, these can be used to estimate tame functions. We demonstrate promising computational results.
Miura surfaces are the solutions of a constrained nonlinear elliptic system of equations. This system is derived by homogenization from the Miura fold, which is a type of origami fold with multiple applications in engineering. A previous inquiry, gave suboptimal conditions for existence of solutions and proposed an $H^2$-conformal finite element method to approximate them. In this paper, the existence of Miura surfaces is studied using a mixed formulation. It is also proved that the constraints propagate from the boundary to the interior of the domain for well-chosen boundary conditions. Then, a numerical method based on a least-squares formulation, Taylor--Hood finite elements and a Newton method is introduced to approximate Miura surfaces. The numerical method is proved to converge and numerical tests are performed to demonstrate its robustness.
In this paper, we study the Radial Basis Function (RBF) approximation to differential operators on smooth tensor fields defined on closed Riemannian submanifolds of Euclidean space, identified by randomly sampled point cloud data. {The formulation in this paper leverages a fundamental fact that the covariant derivative on a submanifold is the projection of the directional derivative in the ambient Euclidean space onto the tangent space of the submanifold. To differentiate a test function (or vector field) on the submanifold with respect to the Euclidean metric, the RBF interpolation is applied to extend the function (or vector field) in the ambient Euclidean space. When the manifolds are unknown, we develop an improved second-order local SVD technique for estimating local tangent spaces on the manifold. When the classical pointwise non-symmetric RBF formulation is used to solve Laplacian eigenvalue problems, we found that while accurate estimation of the leading spectra can be obtained with large enough data, such an approximation often produces irrelevant complex-valued spectra (or pollution) as the true spectra are real-valued and positive. To avoid such an issue,} we introduce a symmetric RBF discrete approximation of the Laplacians induced by a weak formulation on appropriate Hilbert spaces. Unlike the non-symmetric approximation, this formulation guarantees non-negative real-valued spectra and the orthogonality of the eigenvectors. Theoretically, we establish the convergence of the eigenpairs of both the Laplace-Beltrami operator and Bochner Laplacian {for the symmetric formulation} in the limit of large data with convergence rates. Numerically, we provide supporting examples for approximations of the Laplace-Beltrami operator and various vector Laplacians, including the Bochner, Hodge, and Lichnerowicz Laplacians.
This article proposes a highly accurate and conservative method for hyperbolic systems using the finite volume approach. This innovative scheme constructs the intermediate states at the interfaces of the control volumes using the method of characteristics. The approach is simple to implement, generates entropic solutions, and avoids solving Riemann problems. A diffusion control parameter is introduced to increase the accuracy of the scheme. Numerical examples are presented for the Euler equation for an ideal gas. The results demonstrate the method's ability to capture contact discontinuity and shock wave profiles with high accuracy and low cost as well as its robustness.
Nonlinear Fokker-Planck equations play a major role in modeling large systems of interacting particles with a proved effectiveness in describing real world phenomena ranging from classical fields such as fluids and plasma to social and biological dynamics. Their mathematical formulation has often to face with physical forces having a significant random component or with particles living in a random environment which characterization may be deduced through experimental data and leading consequently to uncertainty-dependent equilibrium states. In this work, to address the problem of effectively solving stochastic Fokker-Planck systems, we will construct a new equilibrium preserving scheme through a micro-macro approach based on stochastic Galerkin methods. The resulting numerical method, contrarily to the direct application of a stochastic Galerkin projection in the parameter space of the unknowns of the underlying Fokker-Planck model, leads to highly accurate description of the uncertainty dependent large time behavior. Several numerical tests in the context of collective behavior for social and life sciences are presented to assess the validity of the present methodology against standard ones.
The paper presents a numerical method for the simulation of flow and mechanics in fractured rock. The governing equations which couple the effects in the rock mass and in the fractures are obtained using the discrete fracture-matrix approach. The fracture flow is driven by the cubic law, and the non-penetration contact conditions prevent fractures from closing. A stable finite element discretization is proposed for the displacement-pressure-flux formulation. The resulting nonlinear algebraic system of equations and inequalities is decoupled using a robust iterative splitting into the linearized flow subproblem, and the quadratic programming problem for the mechanical part. The non-penetration conditions are solved by means of the MPGP algorithm. The capability of the numerical scheme is demonstrated on a benchmark problem for borehole excavation with hundreds of fractures in 3D. The paper's novelty consists in combination of three crucial ingredients: (i) application of discrete fracture matrix approach, (ii) robust iterative splitting of resulting nonlinear algebraic system working for real-world 3D problems and (iii) efficient solution of its mechanical quadratic programming part with large number of fractures in mutual contact by means of own solvers with known rate of convergence implemented into in-house PERMON library.
DPPs were introduced by Macchi as a model in quantum optics the 1970s. Since then, they have been widely used as models and subsampling tools in statistics and computer science. Most applications require sampling from a DPP, and given their quantum origin, it is natural to wonder whether sampling a DPP on a quantum computer is easier than on a classical one. We focus here on DPPs over a finite state space, which are distributions over the subsets of $\{1,\dots,N\}$ parametrized by an $N\times N$ Hermitian kernel matrix. Vanilla sampling consists in two steps, of respective costs $\mathcal{O}(N^3)$ and $\mathcal{O}(Nr^2)$ operations on a classical computer, where $r$ is the rank of the kernel matrix. A large first part of the current paper consists in explaining why the state-of-the-art in quantum simulation of fermionic systems already yields quantum DPP sampling algorithms. We then modify existing quantum circuits, and discuss their insertion in a full DPP sampling pipeline that starts from practical kernel specifications. The bottom line is that, with $P$ (classical) parallel processors, we can divide the preprocessing cost by $P$ and build a quantum circuit with $\mathcal{O}(Nr)$ gates that sample a given DPP, with depth varying from $\mathcal{O}(N)$ to $\mathcal{O}(r\log N)$ depending on qubit-communication constraints on the target machine. We also connect existing work on the simulation of superconductors to Pfaffian point processes, which generalize DPPs and would be a natural addition to the machine learner's toolbox. In particular, we describe "projective" Pfaffian point processes, the cardinality of which has constant parity, almost surely. Finally, the circuits are empirically validated on a classical simulator and on 5-qubit IBM machines.
An enriched hybrid high-order method is designed for the Stokes equations of fluid flow and is fully applicable to generic curved meshes. Minimal regularity requirements of the enrichment spaces are given, and an abstract error analysis of the scheme is provided. The method achieves consistency in the enrichment space and is proven to converge optimally in energy error. The scheme is applied to 2D flow around circular cylinders, for which the local behaviour of the velocity and pressure fields are known. By enriching the local spaces with these solutions, superior numerical results near the submerged cylinders are achieved.
The weak Galerkin (WG) finite element method has shown great potential in solving various type of partial differential equations. In this paper, we propose an arbitrary order locking-free WG method for solving linear elasticity problems, with the aid of an appropriate $H(div)$-conforming displacement reconstruction operator. Optimal order locking-free error estimates in both the $H^1$-norm and the $L^2$-norm are proved, i.e., the error is independent of the $Lam\acute{e}$ constant $\lambda$. Moreover, the term $\lambda\|\nabla\cdot \mathbf{u}\|_k$ does not need to be bounded in order to achieve these estimates. We validate the accuracy and the robustness of the proposed locking-free WG algorithm by numerical experiments.
We present the numerical analysis of a finite element method (FEM) for one-dimensional Dirichlet problems involving the logarithmic Laplacian (the pseudo-differential operator that appears as a first-order expansion of the fractional Laplacian as the exponent $s\to 0^+$). Our analysis exhibits new phenomena in this setting; in particular, using recently obtained regularity results, we prove rigorous error estimates and provide a logarithmic order of convergence in the energy norm using suitable \emph{log}-weighted spaces. Numerical evidence suggests that this type of rate cannot be improved. Moreover, we show that the stiffness matrix of logarithmic problems can be obtained as the derivative of the fractional stiffness matrix evaluated at $s=0$. Lastly, we investigate the relationship between the discrete eigenvalue problem and its convergence to the continuous one.
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