In this paper, we propose a new formulation and a suitable finite element method for the steady coupling of viscous flow in deformable porous media using divergence-conforming filtration fluxes. The proposed method is based on the use of parameter-weighted spaces, which allows for a more accurate and robust analysis of the continuous and discrete problems. Furthermore, we conduct a solvability analysis of the proposed method and derive optimal error estimates in appropriate norms. These error estimates are shown to be robust in the case of large Lam\'e parameters and small permeability and storativity coefficients. To illustrate the effectiveness of the proposed method, we provide a few representative numerical examples, including convergence verification, poroelastic channel flow simulation, and test the robustness of block-diagonal preconditioners with respect to model parameters.
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In this paper, we define and study variants of several complexity classes of decision problems that are defined via some criteria on the number of accepting paths of an NPTM. In these variants, we modify the acceptance criteria so that they concern the total number of computation paths instead of the number of accepting ones. This direction reflects the relationship between the counting classes #P and TotP, which are the classes of functions that count the number of accepting paths and the total number of paths of NPTMs, respectively. The former is the well-studied class of counting versions of NP problems introduced by Valiant (1979). The latter contains all self-reducible counting problems in #P whose decision version is in P, among them prominent #P-complete problems such as Non-negative Permanent, #PerfMatch, and #DNF-Sat, thus playing a significant role in the study of approximable counting problems. We show that almost all classes introduced in this work coincide with their `#accepting paths'-definable counterparts, thus providing an alternative model of computation for them. Moreover, for each of these classes, we present a novel family of complete problems, which are defined via TotP-complete problems. This way, we show that all the aforementioned classes have complete problems that are defined via counting problems whose existence version is in P, in contrast to the standard way of obtaining completeness results via counting versions of NP-complete problems. To the best of our knowledge, prior to this work, such results were known only for parity-P and C=P.
In this paper we introduce an abstract setting for the convergence analysis of the virtual element approximation of an acoustic vibration problem. We discuss the effect of the stabilization parameters and remark that in some cases it is possible to achieve optimal convergence without the need of any stabilization. This statement is rigorously proved for lowest order triangular element and supported by several numerical experiments.
In this paper, we propose a new efficient method for calculating the Gerber-Shiu discounted penalty function. Generally, the Gerber-Shiu function usually satisfies a class of integro-differential equation. We introduce the physics-informed neural networks (PINN) which embed a differential equation into the loss of the neural network using automatic differentiation. In addition, PINN is more free to set boundary conditions and does not rely on the determination of the initial value. This gives us an idea to calculate more general Gerber-Shiu functions. Numerical examples are provided to illustrate the very good performance of our approximation.
In this paper, we are concerned about the lattice Boltzmann methods (LBMs) based on vector-kinetic models for hyperbolic partial differential equations. In addition to usual lattice Boltzmann equation (LBE) derived by explicit discretisation of vector-kinetic equation (VKE), we also consider LBE derived by semi-implicit discretisation of VKE and compare the relaxation factors of both. We study the properties such as H-inequality, total variation boundedness and positivity of both the LBEs, and infer that the LBE due to semi-implicit discretisation naturally satisfies all the properties while the LBE due to explicit discretisation requires more restrictive condition on relaxation factor compared to the usual condition obtained from Chapman-Enskog expansion. We also derive the macroscopic finite difference form of the LBEs, and utilise it to establish the consistency of LBEs with the hyperbolic system. Further, we extend this LBM framework to hyperbolic conservation laws with source terms, such that there is no spurious numerical convection due to imbalance between convection and source terms. We also present a D$2$Q$9$ model that allows upwinding even along diagonal directions in addition to the usual upwinding along coordinate directions. The different aspects of the results are validated numerically on standard benchmark problems.
In this paper we study the convergence of a second order finite volume approximation of the scalar conservation law. This scheme is based on the generalized Riemann problem (GRP) solver. We firstly investigate the stability of the GRP scheme and find that it might be entropy unstable when the shock wave is generated. By adding an artificial viscosity we propose a new stabilized GRP scheme. Under the assumption that numerical solutions are uniformly bounded, we prove consistency and convergence of this new GRP method.
This paper explores the application of the multiscale finite element method (MsFEM) to address steady-state Stokes-Darcy problems with BJS interface conditions in highly heterogeneous porous media. We assume the existence of multiscale features in the Darcy region and propose an algorithm for the multiscale Stokes-Darcy model. During the offline phase, we employ MsFEM to construct permeability-dependent offline bases for efficient coarse-grid simulation, with this process conducted in parallel to enhance its efficiency. In the online phase, we use the Robin-Robin algorithm to derive the model's solution. Subsequently, we conduct error analysis based on $L^2$ and $H^1$ norms, assuming certain periodic coefficients in the Darcy region. To validate our approach, we present extensive numerical tests on highly heterogeneous media, illustrating the results of the error analysis.
In this paper, the surface of revolution discrete element method (SR-DEM) is introduced to simulate systems of particles with closed surfaces of revolution. Due to the cylindrical symmetry of a surface of revolution, the geometry of any cross-section about the axis of rotation remains the same. Taking advantage of this geometric feature, a node-to-cross-section contact algorithm is proposed for efficient contact detection between particles with a surface of revolution. In our SR-DEM framework, the contact algorithm is realized in a master-slave fashion: the master particle is approximated by its surface nodes, while the slave particle is represented by a signed distance field (SDF) of the cross-section about the axis of rotation. This hybrid formulation in both 2D and 3D space allows a very efficient contact calculation yet relatively simple code implementation. We then apply SR-DEM to simulate particle-particle, particle-wall impact, granular packing in a cylindrical container, and tablets in a rotating drum, to demonstrate SR-DEM's ability to predict the post-impact velocities, packing porosity, and dynamic angle of repose, respectively. Finally, we suggest a simple approach to find an optimal surface resolution, by increasing the number of surface nodes until some of the bulk properties that could characterize the system converge.
In the present paper, we propose a block variant of the extended Hessenberg process for computing approximations of matrix functions and other problems producing large-scale matrices. Applications to the computation of a matrix function such as f(A)V, where A is an nxn large sparse matrix, V is an nxp block with p<<n, and f is a function are presented. Solving shifted linear systems with multiple right hand sides are also given. Computing approximations of these matrix problems appear in many scientific and engineering applications. Different numerical experiments are provided to show the effectiveness of the proposed method for these problems.
By using the stochastic particle method, the truncated Euler-Maruyama (TEM) method is proposed for numerically solving McKean-Vlasov stochastic differential equations (MV-SDEs), possibly with both drift and diffusion coefficients having super-linear growth in the state variable. Firstly, the result of the propagation of chaos in the L^q (q\geq 2) sense is obtained under general assumptions. Then, the standard 1/2-order strong convergence rate in the L^q sense of the proposed method corresponding to the particle system is derived by utilizing the stopping time analysis technique. Furthermore, long-time dynamical properties of MV-SDEs, including the moment boundedness, stability, and the existence and uniqueness of the invariant probability measure, can be numerically realized by the TEM method. Additionally, it is proven that the numerical invariant measure converges to the underlying one of MV-SDEs in the L^2-Wasserstein metric. Finally, the conclusions obtained in this paper are verified through examples and numerical simulations.
This paper explores an iterative coupling approach to solve linear thermo-poroelasticity problems, with its application as a high-fidelity discretization utilizing finite elements during the training of projection-based reduced order models. One of the main challenges in addressing coupled multi-physics problems is the complexity and computational expenses involved. In this study, we introduce a decoupled iterative solution approach, integrated with reduced order modeling, aimed at augmenting the efficiency of the computational algorithm. The iterative coupling technique we employ builds upon the established fixed-stress splitting scheme that has been extensively investigated for Biot's poroelasticity. By leveraging solutions derived from this coupled iterative scheme, the reduced order model employs an additional Galerkin projection onto a reduced basis space formed by a small number of modes obtained through proper orthogonal decomposition. The effectiveness of the proposed algorithm is demonstrated through numerical experiments, showcasing its computational prowess.
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