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Abstract: Abstract In this paper, we introduce a method for computing rigorous local inclusions of solutions of Cauchy problems for nonlinear heat equations for complex time values. The proof is constructive and provides explicit bounds for the inclusion of the solution of the Cauchy problem, which is rewritten as a zero-finding problem on a certain Banach space. Using a solution map operator, we construct a simplified Newton operator and show that it has a unique fixed point. The fixed point together with its rigorous bounds provides the local inclusion of the solution of the Cauchy problem. The local inclusion technique is then applied iteratively to compute solutions over long time intervals. This technique is used to prove the existence of a branching singularity in the nonlinear heat equation. Finally, we introduce an approach based on the Lyapunov–Perron method for calculating part of a center-stable manifold and prove that an open set of solutions of the Cauchy problem converge to zero, hence yielding the global existence of the solutions in the complex plane of time. PubDate: 2022-05-12
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Abstract: Abstract Inspired by the successes of stochastic algorithms in the training of deep neural networks and the simulation of interacting particle systems, we propose and analyze a framework for randomized time-splitting in linear-quadratic optimal control. In our proposed framework, the linear dynamics of the original problem is replaced by a randomized dynamics. To obtain the randomized dynamics, the system matrix is split into simpler submatrices and the time interval of interest is split into subintervals. The randomized dynamics is then found by selecting randomly one or more submatrices in each subinterval. We show that the dynamics, the minimal values of the cost functional, and the optimal control obtained with the proposed randomized time-splitting method converge in expectation to their analogues in the original problem when the time grid is refined. The derived convergence rates are validated in several numerical experiments. Our numerical results also indicate that the proposed method can lead to a reduction in computational cost for the simulation and optimal control of large-scale linear dynamical systems. PubDate: 2022-05-11
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Abstract: Abstract We propose and analyze new numerical methods to evaluate fractional norms and apply fractional powers of elliptic operators. By means of a reduced basis method, we project to a small dimensional subspace where explicit diagonalization via the eigensystem is feasible. The method relies on several independent evaluations of \(({{\,\mathrm{I}\,}}-t_i^2\Delta )^{-1}f\) , which can be computed in parallel. We prove exponential convergence rates for the optimal choice of sampling points \(t_i\) , provided by the so-called Zolotarëv points. Numerical experiments confirm the analysis and demonstrate the efficiency of our algorithm. PubDate: 2022-05-07
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Abstract: In this paper we construct Ritz-type projectors with boundary interpolation properties in finite dimensional subspaces of the usual Sobolev space and we provide a priori error estimates for them. The abstract analysis is exemplified by considering spline spaces and we equip the corresponding error estimates with explicit constants. This complements our results recently obtained for explicit spline error estimates based on the classical Ritz projectors in (Numer Math 144(4):889–929, 2020). PubDate: 2022-05-03
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Abstract: Abstract This paper proposes two convergent adaptive mesh-refining algorithms for the hybrid high-order method in convex minimization problems with two-sided p-growth. Examples include the p-Laplacian, an optimal design problem in topology optimization, and the convexified double-well problem. The hybrid high-order method utilizes a gradient reconstruction in the space of piecewise Raviart–Thomas finite element functions without stabilization on triangulations into simplices or in the space of piecewise polynomials with stabilization on polytopal meshes. The main results imply the convergence of the energy and, under further convexity properties, of the approximations of the primal resp. dual variable. Numerical experiments illustrate an efficient approximation of singular minimizers and improved convergence rates for higher polynomial degrees. Computer simulations provide striking numerical evidence that an adopted adaptive HHO algorithm can overcome the Lavrentiev gap phenomenon even with empirical higher convergence rates. PubDate: 2022-04-19
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Abstract: Abstract This work concerns the numerical approximation with an explicit first-order finite volume method of inviscid, nonequilibrium, high-temperature flows in multiple space dimensions. It is devoted to the analysis of the numerical scheme for the approximation of the hyperbolic system in homogeneous form. We derive a general framework for the design of numerical schemes for this model from numerical schemes for the monocomponent compressible Euler equations for a polytropic gas. Under a very simple condition on the adiabatic exponent of the polytropic gas, the scheme for the multicomponent system enjoys the same properties as the one for the monocomponent system: discrete entropy inequality, positivity of the partial densities and internal energies, discrete maximum principle on the mass fractions, and discrete minimum principle on the entropy. Our approach extends the relaxation of energy (Coquel and Perthame in SIAM J. Numer. Anal. 35:2223–2249, 1998) to the multicomponent Euler system. In the limit of instantaneous relaxation we show that the solution formally converges to a unique and stable equilibrium solution to the multicomponent Euler equations. We then use this framework to design numerical schemes from three schemes for the polytropic Euler system: the Godunov exact Riemann solver, and the HLL and pressure relaxation based approximate Riemann solvers. Numerical experiments in one and two space dimensions on flows with discontinuous solutions support the conclusions of our analysis and highlight stability, robustness and convergence of the schemes. PubDate: 2022-04-11
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Abstract: Abstract We construct the exponential map associated to a nonholonomic system that allows us to define an exact discrete nonholonomic constraint submanifold. We reproduce the continuous nonholonomic flow as a discrete flow on this discrete constraint submanifold deriving an exact discrete version of the nonholonomic equations. Finally, we derive a general family of nonholonomic integrators that includes as a particular case the exact discrete nonholonomic trajectory. PubDate: 2022-04-11
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Abstract: Abstract In this paper, we consider a drift-diffusion system with cross-coupling through the chemical potentials comprising a model for the motion of finite size ions in liquid electrolytes. The drift term is due to the self-consistent electric field maintained by the ions and described by a Poisson equation. We design two finite volume schemes based on different formulations of the fluxes. We also provide a stability analysis of these schemes and an existence result for the corresponding discrete solutions. A convergence proof is proposed for non-degenerate solutions. Numerical experiments show the behavior of these schemes. PubDate: 2022-04-09
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Abstract: Abstract In this paper, we consider a non-linear fourth-order evolution equation of Cahn–Hilliard-type on evolving surfaces with prescribed velocity, where the non-linear terms are only assumed to have locally Lipschitz derivatives. High-order evolving surface finite elements are used to discretise the weak equation system in space, and a modified matrix–vector formulation for the semi-discrete problem is derived. The anti-symmetric structure of the equation system is preserved by the spatial discretisation. A new stability proof, based on this structure, combined with consistency bounds proves optimal-order and uniform-in-time error estimates. The paper is concluded by a variety of numerical experiments. PubDate: 2022-04-05
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Abstract: Abstract For a well-posed non-selfadjoint indefinite second-order linear elliptic PDE with general coefficients \({\mathbf {A}}, {\mathbf {b}},\gamma \) in \(L^\infty \) and symmetric and uniformly positive definite coefficient matrix \({\mathbf {A}}\) , this paper proves that mixed finite element problems are uniquely solvable and the discrete solutions are uniformly bounded, whenever the underlying shape-regular triangulation is sufficiently fine. This applies to the Raviart-Thomas and Brezzi-Douglas-Marini finite element families of any order and in any space dimension and leads to the best-approximation estimate in \(H({{\,\mathrm{div}\,}})\times L^2\) as well as in in \(L^2\times L^2\) up to oscillations. This generalises earlier contributions for piecewise Lipschitz continuous coefficients to \(L^\infty \) coefficients. The compactness argument of Schatz and Wang for the displacement-oriented problem does not apply immediately to the mixed formulation in \(H({{\,\mathrm{div}\,}})\times L^2\) . But it allows the uniform approximation of some \(L^2\) contributions and can be combined with a recent \(L^2\) best-approximation result from the medius analysis. This technique circumvents any regularity assumption and the application of a Fortin interpolation operator. PubDate: 2022-04-01
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Abstract: Abstract In this paper we introduce a numerical scheme for fluid–structure interaction problems in two or three space dimensions. A flexible elastic plate is interacting with a viscous, compressible barotropic fluid. Hence the physical domain of definition (the domain of Eulerian coordinates) is changing in time. We introduce a fully discrete scheme that is stable, satisfies geometric conservation, mass conservation and the positivity of the density. We also prove that the scheme is consistent with the definition of continuous weak solutions. PubDate: 2022-03-31
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Abstract: Abstract The estimation of probability density functions (PDF) using approximate maps is a fundamental building block in computational probability. We consider forward problems in uncertainty quantification: the inputs or the parameters of a deterministic model are random with a known distribution. The scalar quantity of interest is a fixed measurable function of the parameters, and is therefore a random variable as well. Often, the quantity of interest map is not explicitly known and difficult to compute. Hence, the computational problem is to design a good approximation (surrogate model) of the quantity of interest. For the goal of approximating the moments of the quantity of interest, there is a well developed body of research. One widely popular approach is generalized polynomial chaos (gPC) and its many variants, which approximate moments with spectral accuracy. However, it is not clear whether the PDF of the quantity of interest can be approximated with spectral accuracy as well. This result does not follow directly from spectrally accurate moment estimation. In this paper, we prove convergence rates for PDFs using collocation and Galerkin gPC methods with Legendre polynomials in all dimensions. In particular, exponential convergence of the densities is guaranteed for analytic quantities of interest. In one dimension, we provide more refined results with stronger convergence rates, as well as an alternative proof strategy based on optimal-transport techniques. PubDate: 2022-03-30
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Abstract: Abstract The scattering of electromagnetic waves from obstacles with wave-material interaction in thin layers on the surface is described by generalized impedance boundary conditions, which provide effective approximate models. In particular, this includes a thin coating around a perfect conductor and the skin effect of a highly conducting material. The approach taken in this work is to derive, analyse and discretize a system of time-dependent boundary integral equations that determines the tangential traces of the scattered electric and magnetic fields. In a familiar second step, the fields are evaluated in the exterior domain by a representation formula, which uses the time-dependent potential operators of Maxwell’s equations. The time-dependent boundary integral equation is discretized with Runge–Kutta based convolution quadrature in time and Raviart–Thomas boundary elements in space. Using the frequency-explicit bounds from the well-posedness analysis given here together with known approximation properties of the numerical methods, the full discretization is proved to be stable and convergent, with explicitly given rates in the case of sufficient regularity. Taking the same Runge–Kutta based convolution quadrature for discretizing the time-dependent representation formulas, the optimal order of convergence is obtained away from the scattering boundary, whereas an order reduction occurs close to the boundary. The theoretical results are illustrated by numerical experiments. PubDate: 2022-03-27
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Abstract: Abstract The classical serendipity and mixed finite element spaces suffer from poor approximation on nondegenerate, convex quadrilaterals. In this paper, we develop families of direct serendipity and direct mixed finite element spaces, which achieve optimal approximation properties and have minimal local dimension. The set of local shape functions for either the serendipity or mixed elements contains the full set of scalar or vector polynomials of degree r, respectively, defined directly on each element (i.e., not mapped from a reference element). Because there are not enough degrees of freedom for global \(H^1\) or \(H(\text {div})\) conformity, exactly two supplemental shape functions must be added to each element when \(r\ge 2\) , and only one when \(r=1\) . The specific choice of supplemental functions gives rise to different families of direct elements. These new spaces are related through a de Rham complex. For index \(r\ge 1\) , the new families of serendipity spaces \({\mathscr {DS}}_{r+1}\) are the precursors under the curl operator of our direct mixed finite element spaces, which can be constructed to have reduced or full \(H(\text {div})\) approximation properties. One choice of direct serendipity supplements gives the precursor of the recently introduced Arbogast–Correa spaces (SIAM J Numer Anal 54:3332–3356, 2016. https://doi.org/10.1137/15M1013705). Other fully direct serendipity supplements can be defined without the use of mappings from reference elements, and these give rise in turn to fully direct mixed spaces. Our development is constructive, so we are able to give global bases for our spaces. Numerical results are presented to illustrate their properties. PubDate: 2022-03-26
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Abstract: Abstract This paper presents a new parameter free partially penalized immersed finite element method and convergence analysis for solving second order elliptic interface problems. A lifting operator is introduced on interface edges to ensure the coercivity of the method without requiring an ad-hoc stabilization parameter. The optimal approximation capabilities of the immersed finite element space is proved via a novel new approach that is much simpler than that in the literature. A new trace inequality which is necessary to prove the optimal convergence of immersed finite element methods is established on interface elements. Optimal error estimates are derived rigorously with the constant independent of the interface location relative to the mesh. The new method and analysis have also been extended to variable coefficients and three-dimensional problems. Numerical examples are also provided to confirm the theoretical analysis and efficiency of the new method. PubDate: 2022-03-24
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Abstract: Abstract In order to construct regularizations of continuous linear functionals acting on Sobolev spaces such as \(W_0^{1,q}(\varOmega )\) , where \(1<q<\infty \) and \(\varOmega \) is a Lipschitz domain, we propose a projection method in negative Sobolev spaces \(W^{-1,p}(\varOmega )\) , p being the conjugate exponent satisfying \(p^{-1} + q^{-1} = 1\) . Our method is particularly useful when one is dealing with a rough (irregular) functional that is a member of \(W^{-1,p}(\varOmega )\) , though not of \(L^1(\varOmega )\) , but one strives for a regular approximation in \(L^1(\varOmega )\) . We focus on projections onto discrete finite element spaces \(G_n\) , and consider both discontinuous as well as continuous piecewise-polynomial approximations. While the proposed method aims to compute the best approximation as measured in the negative (dual) norm, for practical reasons, we will employ a computable, discrete dual norm that supremizes over a discrete subspace \(V_m\) . We show that this idea leads to a fully discrete method given by a mixed problem on \(V_m\times G_n\) . We propose a discontinuous as well as a continuous lowest-order pair, prove that they are compatible, and therefore obtain quasi-optimally convergent methods . We present numerical experiments that compute finite element approximations to Dirac delta’s and line sources. We also present adaptively generate meshes, obtained from an error representation that comes with the method. Finally, we show how the presented projection method can be used to efficiently compute numerical approximations to partial differential equations with rough data. PubDate: 2022-03-21
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Abstract: Abstract Consider the scattering of an incident wave by a rigid obstacle, which is immersed in a homogeneous and isotropic elastic medium in two dimensions. Based on a Dirichlet-to-Neumann (DtN) operator, an exact transparent boundary condition is introduced and the scattering problem is formulated as a boundary value problem of the elastic wave equation in a bounded domain. By developing a new duality argument, an a posteriori error estimate is derived for the discrete problem by using the finite element method with the truncated DtN operator. The a posteriori error estimate consists of the finite element approximation error and the truncation error of the DtN operator, where the latter decays exponentially with respect to the truncation parameter. An adaptive finite element algorithm is proposed to solve the elastic obstacle scattering problem, where the truncation parameter is determined through the truncation error and the mesh elements for local refinements are chosen through the finite element discretization error. Numerical experiments are presented to demonstrate the effectiveness of the proposed method. PubDate: 2022-03-19
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Abstract: Abstract We study the Stokes problem over convex polyhedral domains on weighted Sobolev spaces. The weight is assumed to belong to the Muckenhoupt class \(\varvec{A}_{\varvec{q}}\) for \(\varvec{q} \in (1,\varvec{\infty })\) . We show that the Stokes problem is well-posed for all \(\varvec{q}\) . In addition, we show that the finite element Stokes projection is stable on weighted spaces. With the aid of these tools, we provide well-posedness and approximation results to some classes of non-Newtonian fluids. PubDate: 2022-03-04 DOI: 10.1007/s00211-022-01272-5
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Abstract: Abstract In this paper we propose and study an implicit finite volume scheme for a general model which describes the evolution of the composition of a multi-component mixture in a bounded domain. We assume that the whole domain is occupied by the different phases of the mixture which leads to a volume filling constraint. In the continuous model this constraint yields the introduction of a pressure, which should be thought as a Lagrange multiplier for the volume filling constraint. The pressure solves an elliptic equation, to be coupled with parabolic equations, possibly including cross-diffusion terms, which govern the evolution of the mixture composition. Besides the system admits an entropy structure which is at the cornerstone of our analysis. More precisely, the main objective of this work is to design a two-point flux approximation finite volume scheme which preserves the key properties of the continuous model, namely the volume filling constraint and the control of the entropy production. Thanks to these properties, and in particular the discrete entropy-entropy dissipation relation, we are able to prove the existence of solutions to the scheme and its convergence. Finally, we illustrate the behavior of our scheme through different applications. PubDate: 2022-02-28 DOI: 10.1007/s00211-022-01270-7
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Abstract: Abstract This paper is concerned with the rigorous error analysis of a fully discrete scheme obtained by using a central fluxes discontinuous Galerkin (dG) method in space and the Peaceman–Rachford splitting scheme in time. We apply the scheme to a general class of wave-type problems and show that the resulting approximations as well as discrete derivatives thereof satisfy error bounds of the order of the polynomial degree used in the dG discretization and order two in time. In particular, the class of problems considered includes, e.g., the advection equation, the acoustic wave equation, and the Maxwell equations for which a very efficient implementation is possible via an alternating direction implicit splitting. PubDate: 2022-01-31 DOI: 10.1007/s00211-021-01262-z
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