Showing papers in "Mathematical Modelling and Numerical Analysis in 2021"
TL;DR: In this paper, a hyperbolic transport model for the propagation in space of an epidemic phenomenon described by a classical compartmental dynamics is proposed, which is based on a kinetic description at discrete velocities of the spatial movement and interactions of a population of susceptible, infected and recovered individuals.
Abstract: We consider the development of hyperbolic transport models for the propagation in space of an epidemic phenomenon described by a classical compartmental dynamics. The model is based on a kinetic description at discrete velocities of the spatial movement and interactions of a population of susceptible, infected and recovered individuals. Thanks to this, the unphysical feature of instantaneous diffusive effects, which is typical of parabolic models, is removed. In particular, we formally show how such reaction-diffusion models are recovered in an appropriate diffusive limit. The kinetic transport model is therefore considered within a spatial network, characterizing different places such as villages, cities, countries, etc. The transmission conditions in the nodes are analyzed and defined. Finally, the model is solved numerically on the network through a finite-volume IMEX method able to maintain the consistency with the diffusive limit without restrictions due to the scaling parameters. Several numerical tests for simple epidemic network structures are reported and confirm the ability of the model to correctly describe the spread of an epidemic.
30 citations
TL;DR: This work proposes a model reduction procedure for rapid and reliable solution of parameterized hyperbolic partial differential equations and presents numerical results for a Burgers model problem and a shallow water model problem, to empirically demonstrate the potential of the method.
Abstract: We propose a model reduction procedure for rapid and reliable solution of parameterized hyperbolic partial differential equations. Due to the presence of parameter-dependent shock waves and contact discontinuities, these problems are extremely challenging for traditional model reduction approaches based on linear approximation spaces. The main ingredients of the proposed approach are (i) an adaptive space-time registration-based data compression procedure to align local features in a fixed reference domain, (ii) a space-time Petrov–Galerkin (minimum residual) formulation for the computation of the mapped solution, and (iii) a hyper-reduction procedure to speed up online computations. We present numerical results for a Burgers model problem and a shallow water model problem, to empirically demonstrate the potential of the method.
23 citations
TL;DR: This work investigates the complexity of a conjugate gradient method applied to the fully discretized OCP, in which the Finite Element discretization and Monte Carlo sample are chosen in advance and kept fixed over the iterations.
Abstract: We consider the numerical approximation of an optimal control problem for an elliptic Partial Differential Equation (PDE) with random coefficients. Specifically, the control function is a deterministic, distributed forcing term that minimizes the expected squared L 2 misfit between the state (i.e. solution to the PDE) and a target function, subject to a regularization for well posedness. For the numerical treatment of this risk-averse Optimal Control Problem (OCP) we consider a Finite Element discretization of the underlying PDE, a Monte Carlo sampling method, and gradient-type iterations to obtain the approximate optimal control. We provide full error and complexity analyses of the proposed numerical schemes. In particular we investigate the complexity of a conjugate gradient method applied to the fully discretized OCP (so called Sample Average Approximation), in which the Finite Element discretization and Monte Carlo sample are chosen in advance and kept fixed over the iterations. This is compared with a Stochastic Gradient method on a fixed or varying Finite Element discretization, in which the expectation in the computation of the steepest descent direction is approximated by Monte Carlo estimators, independent across iterations, with small sample sizes. We show in particular that the second strategy results in an improved computational complexity. The theoretical error estimates and complexity results are confirmed by numerical experiments.
20 citations
TL;DR: In this article, a finite element numerical scheme for the Cahn-Hilliard phase-field model of the two-phase incompressible flow system with variable density and viscosity is presented.
Abstract: We construct a fully-discrete finite element numerical scheme for the Cahn–Hilliard phase-field model of the two-phase incompressible flow system with variable density and viscosity. The scheme is linear, decoupled, and unconditionally energy stable. Its key idea is to combine the penalty method of the Navier–Stokes equations with the Strang operator splitting method, and introduce several nonlocal variables and their ordinary differential equations to process coupled nonlinear terms. The scheme is highly efficient and it only needs to solve a series of completely independent linear elliptic equations at each time step, in which the Cahn–Hilliard equation and the pressure Poisson equation only have constant coefficients. We rigorously prove the unconditional energy stability and solvability of the scheme and carry out numerous accuracy/stability examples and various benchmark numerical simulations in 2D and 3D, including the Rayleigh–Taylor instability and rising/coalescence dynamics of bubbles to demonstrate the effectiveness of the scheme, numerically.
17 citations
TL;DR: Using the Multi-Trace Formalism (MTF), a new variant of the Optimized Schwarz Method (OSM) that remains valid in the presence of cross-points in the subdomain partition is proposed that leads to the derivation of a strongly coercive formulation of the Helmholtz problem posed on the union of all interfaces.
Abstract: We consider a scalar wave propagation in harmonic regime modelled by Helmholtz equation with heterogeneous coefficients. Using the Multi-Trace Formalism (MTF), we propose a new variant of the Optimized Schwarz Method (OSM) that remains valid in the presence of cross-points in the subdomain partition. This leads to the derivation of a strongly coercive formulation of our Helmholtz problem posed on the union of all interfaces. The corresponding operator takes the form “identity + non-expansive”.
16 citations
TL;DR: It is shown that some of the proposed error estimators perform better than or equally well as the existing ones and can be easily extended to estimate the output error of reduced-order modeling for steady linear parametric systems.
Abstract: Motivated by a recently proposed error estimator for the transfer function of the reduced-order model of a given linear dynamical system, we further develop more theoretical results in this work. Moreover, we propose several variants of the error estimator, and compare those variants with the existing ones both theoretically and numerically. It is shown that some of the proposed error estimators perform better than or equally well as the existing ones. All the error estimators considered can be easily extended to estimate the output error of reduced-order modeling for steady linear parametric systems.
15 citations
TL;DR: A residual-based a posteriori error estimator for reduced models learned with non-intrusive model reduction from data of high-dimensional systems governed by linear parabolic partial differential equations with control inputs is derived.
Abstract: This work derives a residual-based a posteriori error estimator for reduced models learned with non-intrusive model reduction from data of high-dimensional systems governed by linear parabolic partial differential equations with control inputs. It is shown that quantities that are necessary for the error estimator can be either obtained exactly as the solutions of least-squares problems in a non-intrusive way from data such as initial conditions, control inputs, and high-dimensional solution trajectories or bounded in a probabilistic sense. The computational procedure follows an offline/online decomposition. In the offline (training) phase, the high-dimensional system is judiciously solved in a black-box fashion to generate data and to set up the error estimator. In the online phase, the estimator is used to bound the error of the reduced-model predictions for new initial conditions and new control inputs without recourse to the high-dimensional system. Numerical results demonstrate the workflow of the proposed approach from data to reduced models to certified predictions.
14 citations
TL;DR: In this article, a detailed analysis of exact and approximate fast, intermediate, and slow modes together with improved estimates for the solutions and their time derivatives, and the time-integration method is presented.
Abstract: Convergence rate estimates are obtained for singular limits of the compressible ideal magnetohydrodynamics equations, in which the Mach and Alfven numbers tend to zero at different rates. The proofs use a detailed analysis of exact and approximate fast, intermediate, and slow modes together
with improved estimates for the solutions and their time derivatives, and the time-integration method. When the small parameters are related by a power law the convergence rates are positive powers of the Mach number, with the power varying depending on the component and the norm. Exceptionally, the convergence rate for two components involve the ratio of the two parameters, and that rate is proven to be sharp via corrector terms. Moreover, the convergence rates for the case of a power-law relation
between the small parameters tend to the two-scale convergence rate as the power tends to one. These
results demonstrate that the issue of convergence rates for three-scale singular limits, which was not
addressed in the authors' previous paper, is much more complicated than for the classical two-scale
singular limits.
14 citations
TL;DR: This work considers a primal-dual discrete formulation of the continuum problem with the addition of stabilization terms that are designed with the goal of minimizing the numerical errors, and proves error estimates using the stability properties of the numerical scheme and a continuum observability estimate based on the sharp geometric control condition.
Abstract: We consider a stabilized finite element method based on a spacetime formulation, where the equations are solved on a global (unstructured) spacetime mesh. A unique continuation problem for the wave equation is considered, where a noisy data is known in an interior subset of spacetime. For this problem, we consider a primal-dual discrete formulation of the continuum problem with the addition of stabilization terms that are designed with the goal of minimizing the numerical errors. We prove error estimates using the stability properties of the numerical scheme and a continuum observability estimate, based on the sharp geometric control condition by Bardos, Lebeau and Rauch. The order of convergence for our numerical scheme is optimal with respect to stability properties of the continuum problem and the approximation order of the finite element residual. Numerical examples are provided that illustrate the methodology.
13 citations
TL;DR: As an application, a multilevel decomposition based on Scott-Zhang operators on a hierarchy of meshes generated by newest vertex bisection with equivalent norms up to (but excluding) the endpoint case is obtained.
Abstract: We provide an endpoint stability result for Scott-Zhang type operators in Besov spaces. For globally continuous piecewise polynomials these are bounded from H 3/2 into ; for element wise polynomials these are bounded from H 1/2 into . As an application, we obtain a multilevel decomposition based on Scott-Zhang operators on a hierarchy of meshes generated by newest vertex bisection with equivalent norms up to (but excluding) the endpoint case. A local multilevel diagonal preconditioner for the fractional Laplacian on locally refined meshes with optimal eigenvalue bounds is presented.
13 citations
TL;DR: In this article, a numerical dissipation switch is proposed to control the amount of dissipation present in central-upwind schemes without risking oscillations, which is achieved with the help of a more accurate estimate of the local propagation speeds in the parts of the computational domain.
Abstract: We propose a numerical dissipation switch, which helps to control the amount of numerical dissipation present in central-upwind schemes. Our main goal is to reduce the numerical dissipation without risking oscillations. This goal is achieved with the help of a more accurate estimate of the local propagation speeds in the parts of the computational domain, which are near contact discontinuities and shears. To this end, we introduce a switch parameter, which depends on the distributions of energy in the x - and y -directions. The resulting new central-upwind is tested on a number of numerical examples, which demonstrate the superiority of the proposed method over the original central-upwind scheme.
TL;DR: In this article, a general higher order operator splitting scheme for diffusion semigroups using the Baker-Campbell-Hausdorff type commutator expansion of non-commutative algebra and the Malliavin calculus is proposed.
Abstract: This paper proposes a general higher order operator splitting scheme for diffusion semigroups using the Baker–Campbell–Hausdorff type commutator expansion of non-commutative algebra and the Malliavin calculus. An accurate discretization method for the fundamental solution of heat equations or the heat kernel is introduced with a new computational algorithm which will be useful for the inference for diffusion processes. The approximation is regarded as the splitting around the Euler–Maruyama scheme for the density. Numerical examples for diffusion processes are shown to validate the proposed scheme.
TL;DR: In this article, the authors developed a numerical scheme for approximating a d-dimensional chemotaxis-Navier-Stokes system, d = 2, 3, modeling cellular swimming in incompressible fluids.
Abstract: In this paper we develop a numerical scheme for approximating a d -dimensional chemotaxis-Navier–Stokes system, d =2, 3, modeling cellular swimming in incompressible fluids This model describes the chemotaxis-fluid interaction in cases where the chemical signal is consumed with a rate proportional to the amount of organisms We construct numerical approximations based on the Finite Element method and analyze optimal error estimates and convergence towards regular solutions In order to construct the numerical scheme, we use a splitting technique to deal with the chemo-attraction term in the cell-density equation, leading to introduce a new variable given by the gradient of the chemical concentration Having the equivalent model, we consider a fully discrete Finite Element approximation which is well-posed and mass-conservative We obtain uniform estimates and analyze the convergence of the scheme Finally, we present some numerical simulations to verify the good behavior of our scheme, as well as to check numerically the optimal error estimates proved in our theoretical analysis
TL;DR: New variants of adaptive Trust-Region methods for parameter optimization with PDE constraints and bilateral parameter constraints are proposed and rigorously analyzed and a non-conforming dual (NCD) approach is proposed to improve the standard RB approximation of the optimality system.
Abstract: In this contribution we propose and rigorously analyze new variants of adaptive Trust-Region methods for parameter optimization with PDE constraints and bilateral parameter constraints. The approach employs successively enriched Reduced Basis surrogate models that are constructed during the outer optimization loop and used as model function for the Trust-Region method. Each Trust-Region sub-problem is solved with the projected BFGS method. Moreover, we propose a non-conforming dual (NCD) approach to improve the standard RB approximation of the optimality system. Rigorous improved a posteriori error bounds are derived and used to prove convergence of the resulting NCD-corrected adaptive Trust-Region Reduced Basis algorithm. Numerical experiments demonstrate that this approach enables to reduce the computational demand for large scale or multi-scale PDE constrained optimization problems significantly.
TL;DR: In this paper, the authors considered the original and modified electromagnetic Steklov eigenvalue problem and constructed a test function operator function T (⋅) so that A(λ ) is weakly T (λ )-coercive for all suitable λ, i.e.
Abstract: We continue the work of Camano et al . [SIAM J. Math. Anal. 49 (2017) 4376–4401] on electromagnetic Steklov eigenvalues. The authors recognized that in general the eigenvalues do not correspond to the spectrum of a compact operator and hence proposed a modified eigenvalue problem with the desired properties. The present article considers the original and the modified electromagnetic Steklov eigenvalue problem. We cast the problems as eigenvalue problem for a holomorphic operator function A (⋅). We construct a “test function operator function” T (⋅) so that A (λ ) is weakly T (λ )-coercive for all suitable λ , i.e. T (λ )*A (λ ) is a compact perturbation of a coercive operator. The construction of T (⋅) relies on a suitable decomposition of the function space into subspaces and an apt sign change on each subspace. For the approximation analysis, we apply the framework of T-compatible Galerkin approximations. For the modified problem, we prove that convenient commuting projection operators imply T-compatibility and hence convergence. For the original problem, we require the projection operators to satisfy an additional commutator property which concerns the tangential trace. The existence and construction of such projection operators remain open questions.
TL;DR: In this paper, the authors prove wellposedness and regularity of solutions to a fractional diffusion porous media equation with a variable fractional order that may depend on the unknown solution and present a linearly implicit time-stepping method to linearize and discretize the equation in time.
Abstract: We prove well-posedness and regularity of solutions to a fractional diffusion porous media equation with a variable fractional order that may depend on the unknown solution. We present a linearly implicit time-stepping method to linearize and discretize the equation in time, and present rigorous analysis for the convergence of numerical solutions based on proved regularity results.
TL;DR: An error estimate is presented for a fully discrete, linearized and stabilized finite element method for solving the coupled system of nonlinear hyperbolic and parabolic equations describing incompressible flow with variable density in a two-dimensional convex polygon.
Abstract: An error estimate is presented for a fully discrete, linearized and stabilized finite element method for solving the coupled system of nonlinear hyperbolic and parabolic equations describing incompressible flow with variable density in a two-dimensional convex polygon. In particular, the error of the numerical solution is split into the temporal and spatial components, separately. The temporal error is estimated by applying discrete maximal L p -regularity of time-dependent Stokes equations, and the spatial error is estimated by using energy techniques based on the uniform regularity of the solutions given by semi-discretization in time.
TL;DR: In this paper, the authors provide a unified analysis of a posteriori and a priori error bounds for a broad class of discontinuous Galerkin and C 0 -IP finite element approximations of fully nonlinear second-order elliptic Hamilton-Jacobi-Bellman and Isaacs equations with Cordes coefficients.
Abstract: We provide a unified analysis of a posteriori and a priori error bounds for a broad class of discontinuous Galerkin and C 0 -IP finite element approximations of fully nonlinear second-order elliptic Hamilton–Jacobi–Bellman and Isaacs equations with Cordes coefficients. We prove the existence and uniqueness of strong solutions in H 2 of Isaacs equations with Cordes coefficients posed on bounded convex domains. We then show the reliability and efficiency of computable residual-based error estimators for piecewise polynomial approximations on simplicial meshes in two and three space dimensions. We introduce an abstract framework for the a priori error analysis of a broad family of numerical methods and prove the quasi-optimality of discrete approximations under three key conditions of Lipschitz continuity, discrete consistency and strong monotonicity of the numerical method. Under these conditions, we also prove convergence of the numerical approximations in the small-mesh limit for minimal regularity solutions. We then show that the framework applies to a range of existing numerical methods from the literature, as well as some original variants. A key ingredient of our results is an original analysis of the stabilization terms. As a corollary, we also obtain a generalization of the discrete Miranda–Talenti inequality to piecewise polynomial vector fields.
TL;DR: This work studies a specific example of library approximation where the parameter domain is split into a finite number N of rectangular cells, with affine spaces of dimension m assigned to each cell, and gives performance guarantees with respect to accuracy of approximation versus m and N.
Abstract: Typical model reduction methods for parametric partial differential equations construct a linear space V n which approximates well the solution manifold M consisting of all solutions u (y ) with y the vector of parameters. In many problems of numerical computation, nonlinear methods such as adaptive approximation, n -term approximation, and certain tree-based methods may provide improved numerical efficiency over linear methods. Nonlinear model reduction methods replace the linear space V n by a nonlinear space Σn . Little is known in terms of their performance guarantees, and most existing numerical experiments use a parameter dimension of at most two. In this work, we make a step towards a more cohesive theory for nonlinear model reduction. Framing these methods in the general setting of library approximation, we give a first comparison of their performance with the performance of standard linear approximation for any compact set. We then study these methods for solution manifolds of parametrized elliptic PDEs. We study a specific example of library approximation where the parameter domain is split into a finite number N of rectangular cells, with affine spaces of dimension m assigned to each cell, and give performance guarantees with respect to accuracy of approximation versus m and N .
TL;DR: Well-posedness of a space-time First-Order System Least-Squares formulation of the heat equation is proven, this result is generalized to general second order parabolic PDEs with possibly inhomogenoeus boundary conditions, and plain convergence of a standard adaptive finite element method driven by the least-squares estimator is demonstrated.
Abstract: In [2019, Space-time least-squares finite elements for parabolic equations, arXiv:1911.01942] by Fuhrer and Karkulik, well-posedness of a space-time First-Order System Least-Squares formulation of the heat equation was proven. In the present work, this result is generalized to general second order parabolic PDEs with possibly inhomogenoeus boundary conditions, and plain convergence of a standard adaptive finite element method driven by the least-squares estimator is demonstrated. The proof of the latter easily extends to a large class of least-squares formulations.
TL;DR: In this article, a nonlinear modification of the right hand side of the system of equations of the spline, that contains divided differences, is proposed to attain adaption close to jumps in the function.
Abstract: When interpolating data with certain regularity, spline functions are useful. They are defined as piecewise polynomials that satisfy certain regularity conditions at the joints. In the literature about splines it is possible to find several references that study the apparition of Gibbs phenomenon close to jump discontinuities in the results obtained by spline interpolation. This work is devoted to the construction and analysis of a new nonlinear technique that allows to improve the accuracy of splines near jump discontinuities eliminating the Gibbs phenomenon. The adaption is easily attained through a nonlinear modification of the right hand side of the system of equations of the spline, that contains divided differences. The modification is based on the use of a new limiter specifically designed to attain adaption close to jumps in the function. The new limiter can be seen as a nonlinear weighted mean that has better adaption properties than the linear weighted mean. We will prove that the nonlinear modification introduced in the spline keeps the maximum theoretical accuracy in all the domain except at the intervals that contain a jump discontinuity, where Gibbs oscillations are eliminated. Diffusion is introduced, but this is fine if the discontinuity appears due to a discretization of a high gradient with not enough accuracy. The new technique is introduced for cubic splines, but the theory presented allows to generalize the results very easily to splines of any order. The experiments presented satisfy the theoretical aspects analyzed in the paper.
TL;DR: In this article, the authors considered a perforated half-cylindrical thin shell and investigated the limit behavior when the period and the thickness simultaneously went to zero, using the decomposition of shell displacements.
Abstract: We consider a perforated half-cylindrical thin shell and investigate the limit behavior when the period and the thickness simultaneously go to zero. By using the decomposition of shell displacements presented in Griso [JMPA 89 (2008) 199–223] we obtain a priori estimates. With the unfolding and rescaling operator we transform the problem to a reference configuration. In the end this yields a homogenized limit problem for the shell.
TL;DR: The LDG method extends the discontinuous Galerkin (DG) method for purely hyperbolic equations to parabolic equations and shares with the DG method its advantage and flexibility, and it is proved the L2-stability of the numerical scheme for fully nonlinear equations.
Abstract: In this paper, we propose a local discontinuous Galerkin (LDG) method for nonlinear and possibly degenerate parabolic stochastic partial differential equations, which is a high-order numerical scheme. It extends the discontinuous Galerkin (DG) method for purely hyperbolic equations to parabolic equations and shares with the DG method its advantage and flexibility. We prove the L 2 -stability of the numerical scheme for fully nonlinear equations. Optimal error estimates (O (h (k+1) )) for smooth solutions of semi-linear stochastic equations is shown if polynomials of degree k are used. We use an explicit derivative-free order 1.5 time discretization scheme to solve the matrix-valued stochastic ordinary differential equations derived from the spatial discretization. Numerical examples are given to display the performance of the LDG method.
TL;DR: The MDFEM is introduced to approximate the expected value of a linear functional of the solution of the PDE which is an infinite-dimensional integral over the parameter space and higher-order convergence rates are achieved in term of error versus cost.
Abstract: We introduce the multivariate decomposition finite element method (MDFEM) for elliptic PDEs with lognormal diffusion coefficients, that is, when the diffusion coefficient has the form a = exp(Z ) where Z is a Gaussian random field defined by an infinite series expansion with and a given sequence of functions . We use the MDFEM to approximate the expected value of a linear functional of the solution of the PDE which is an infinite-dimensional integral over the parameter space. The proposed algorithm uses the multivariate decomposition method (MDM) to compute the infinite-dimensional integral by a decomposition into finite-dimensional integrals, which we resolve using quasi-Monte Carlo (QMC) methods, and for which we use the finite element method (FEM) to solve different instances of the PDE. We develop higher-order quasi-Monte Carlo rules for integration over the finite-dimensional Euclidean space with respect to the Gaussian distribution by use of a truncation strategy. By linear transformations of interlaced polynomial lattice rules from the unit cube to a multivariate box of the Euclidean space we achieve higher-order convergence rates for functions belonging to a class of anchored Gaussian Sobolev spaces while taking into account the truncation error. These cubature rules are then used in the MDFEM algorithm. Under appropriate conditions, the MDFEM achieves higher-order convergence rates in terms of error versus cost, i.e. , to achieve an accuracy of O (e ) the computational cost is where e −1/λ and e −d ′/λ ) are respectively the cost of the quasi-Monte Carlo cubature and the finite element approximations, with d ′= d (1+δ ′) for some δ ′ ≥ 0 and d the physical dimension, and is a parameter representing the sparsity of .
TL;DR: In this article, two families of generalised Hermite polynomials/functions (GHPs/GHFs) in arbitrary dimensions were introduced, and efficient and accurate spectral methods for solving PDEs with integral fractional Laplacian (IFL) and/or Schrodinger operators in ℝ d.
Abstract: In this paper, we introduce two families of nontensorial generalised Hermite polynomials/functions (GHPs/GHFs) in arbitrary dimensions, and develop efficient and accurate spectral methods for solving PDEs with integral fractional Laplacian (IFL) and/or Schrodinger operators in ℝ d . As a generalisation of the G. Szego’s family in 1D (1939), the first family of multivariate GHPs (resp. GHFs) are orthogonal with respect to the weight function (resp. ) in ℝd . We further construct the adjoint generalised Hermite functions (A-GHFs), which have an interwoven connection with the corresponding GHFs through the Fourier transform, and are orthogonal with respect to the inner product associated with the IFL of order s > 0. As an immediate consequence, the spectral-Galerkin method using A-GHFs as basis functions leads to a diagonal stiffness matrix for the IFL (which is known to be notoriously difficult and expensive to discretise). The new basis also finds remarkably efficient in solving PDEs with the fractional Schrodinger operator: with s ∈ (0,1] and μ > −1/2 in ℝ d We construct the second family of multivariate nontensorial Muntz-type GHFs, which are orthogonal with respect to an inner product associated with the underlying Schrodinger operator, and are tailored to the singularity of the solution at the origin. We demonstrate that the Muntz-type GHF spectral method leads to sparse matrices and spectrally accurate solution to some Schrodinger eigenvalue problems.
TL;DR: In this paper, the authors studied the asymptotic behavior of 3D magnetostatics with respect to a singular perturbation of the differential operator and proved the existence of the topological derivative using a Lagrangian approach.
Abstract: In this paper we study the asymptotic behaviour of the quasilinear $\curl$-$\curl$ equation of 3D magnetostatics with respect to a singular perturbation of the differential operator and prove the existence of the topological derivative using a Lagrangian approach. We follow the strategy proposed in our recent previous work ( https://doi.org/10.1051/cocv/2020035 ) where a systematic and concise way for the derivation of topological derivatives for quasi-linear elliptic problems in $H^1$ is introduced. In order to prove the asymptotics for the state equation we make use of an appropriate Helmholtz decomposition. The evaluation of the topological derivative at any spatial point requires the solution of a nonlinear transmission problem. We discuss an efficient way for the numerical evaluation of the topological derivative in the whole design domain using precomputation in an offline stage. This allows us to use the topological derivative for the design optimization of an electrical machine.
TL;DR: In this article, a Yee-like scheme for wave propagation problems on a Minkowski manifold has been proposed, where the constitutive laws are imposed on a finite set of points instead of on all ordinary points of space.
Abstract: Finite difference kind of schemes are popular in approximating wave propagation problems in finite dimensional spaces. While Yee’s original paper on the finite difference method is already from the sixties, mathematically there still remains questions which are not yet satisfactorily covered. In this paper, we address two issues of this kind. Firstly, in the literature Yee’s scheme is constructed separately for each particular type of wave problem. Here, we explicitly generalize the Yee scheme to a class of wave problems that covers at large physics field theories. For this we introduce Yee’s scheme for all problems of a class characterised on a Minkowski manifold by (i) a pair of first order partial differential equations and by (ii) a constitutive relation that couple the differential equations with a Hodge relation. In addition, we introduce a strategy to systematically exploit higher order Whitney elements in Yee-like approaches. This makes higher order interpolation possible both in time and space. For this, we show that Yee-like schemes preserve the local character of the Hodge relation, which is to say, the constitutive laws become imposed on a finite set of points instead of on all ordinary points of space. As a result, the usage of higher order Whitney forms does not compel to change the actual solution process at all. This is demonstrated with a simple example.
TL;DR: In this paper, the authors studied the nonlinear multi-species Boltzmann equation with random uncertainty coming from the initial data and collision kernel, and obtained well-posedness and long-time behavior.
Abstract: In this paper the nonlinear multi-species Boltzmann equation with random uncertainty coming from the initial data and collision kernel is studied. Well-posedness and long-time behavior – exponential decay to the global equilibrium – of the analytical solution, and spectral gap estimate for the corresponding linearized gPC-based stochastic Galerkin system are obtained, by using and extending the analytical tools provided in [M. Briant and E.S. Daus, Arch. Ration. Mech. Anal. 3 (2016) 1367–1443] for the deterministic problem in the perturbative regime, and in [E.S. Daus, S. Jin and L. Liu, Kinet. Relat. Models 12 (2019) 909–922] for the single-species problem with uncertainty. The well-posedness result of the sensitivity system presented here has not been obtained so far neither in the single species case nor in the multi-species case.
TL;DR: Convergence of optimal order is proven in $L_2(H^1)$-norm, independent of the derivatives of the coefficients, in the generalized finite element method for the strongly damped wave equation with highly varying coefficients.
Abstract: We propose a generalized finite element method for the strongly damped wave equation with highly varying coefficients The proposed method is based on the localized orthogonal decomposition introduced in Malqvist and Peterseim [Math Comp 83 (2014) 2583–2603], and is designed to handle independent variations in both the damping and the wave propagation speed respectively The method does so by automatically correcting for the damping in the transient phase and for the propagation speed in the steady state phase Convergence of optimal order is proven in L2 (H1 )-norm, independent of the derivatives of the coefficients We present numerical examples that confirm the theoretical findings
TL;DR: It is shown that, if some regularity properties of the solution are satisfied and if the time step verifies a stability condition, then the family of proposed time discretisations provides, in a strong norm, second order space-time convergence.
Abstract: In this work we present and analyse a time discretisation strategy for linear wave equations t hat aims at using locally in space the most adapted time discretisation among a family of implicit or explicit centered second order schemes. The proposed family of schemes is adapted to domain decomposition methods such as the mortar element method. They correspond in that case to local implicit schemes and to local time stepping. We show that, if some regularity properties of the solution are satisfied and if the time step verifies a stability condition, then the family of proposed time discretisations provides, in a strong norm, second order space-time convergence. Finally, we provide 1D and 2D numerical illustrations that confirm the obtained theoretical results and we compare our approach on 1D test cases to other existing local time stepping strategies for wave equations.