Optimal local approximation spaces for parabolic problems (original) (raw)

A Multiscale Approach for Optimal Control Problems of Linear Parabolic Equations

SIAM Journal on Control and Optimization, 2012

This paper discusses multiscale analysis for optimal control problems of linear parabolic equations with rapidly oscillating coefficients that depend on spatial and temporal variables. There are mainly three new results in the present paper. First, we obtain the convergence results with an explicit convergence rate for the multiscale asymptotic expansions of the solution of the optimal control problem in the case without constraints. Second, for a general bounded Lipschitz polygonal domain, the boundary layer solution is defined and the corresponding convergence results are also derived. Finally, an explicit convergence rate ε 1/2 in the presence of constraint is reported.

Multiscale Galerkin approximation scheme for a system of quasilinear parabolic equations

We discuss a multiscale Galerkin approximation scheme for a system of coupled quasi-linear parabolic equations. These equations arise from the upscaling of a pore scale filtration combustion model under the assumptions of large Damkhöler number and small Péclet number. The upscaled model consists of a heat diffusion equation and a mass diffusion equation in the bulk of a macroscopic domain. The associated diffusion tensors are bivariate functions of temperature and concentration and provide the necessary coupling conditions to elliptic-type cell problems. These cell problems are characterized by a reaction-diffusion phenomenon with nonlinear reactions of Arrhenius type at a gas-solid interface. We discuss the wellposedness of the quasilinear system and establish uniform estimates for the finite dimensional approximations. Based on these estimates, the convergence of the approximating sequence is proved. The results of numerical simulations demonstrate, in suitable temperature regimes, the potential of solutions of the upscaled model to mimic those from porous media combustion. Moreover, distinctions are made between the effects of the microscopic reaction-diffusion processes on the macroscopic system of equations and a purely diffusion system.

Optimal Local Approximation Spaces for Generalized Finite Element Methods with Application to Multiscale Problems

Multiscale Modeling & Simulation, 2011

The paper addresses a numerical method for solving second order elliptic partial differential equations that describe fields inside heterogeneous media. The scope is general and treats the case of rough coefficients, i.e. coefficients with values in L ∞ (Ω). This class of coefficients includes as examples media with micro-structure as well as media with multiple non-separated length scales. The approach taken here is based on the the generalized finite element method (GFEM) introduced in [5], and elaborated in [3], [4] and [25]. The GFEM is constructed by partitioning the computational domain Ω into a collection of preselected subsets ωi, i = 1, 2, ..m and constructing finite dimensional approximation spaces Ψi over each subset using local information. The notion of the Kolmogorov n-width is used to identify the optimal local approximation spaces. These spaces deliver local approximations with errors that decay almost exponentially with the degrees of freedom Ni in the energy norm over ωi. The local spaces Ψi are used within the GFEM scheme to produce a finite dimensional subspace S N of H 1 (Ω) which is then employed in the Galerkin method. It is shown that the error in the Galerkin approximation decays in the energy norm almost exponentially (i.e., super-algebraicly) with respect to the degrees of freedom N. When length scales "separate" and the microstructure is sufficiently fine with respect to the length scale of the domain ωi it is shown that homogenization theory can be used to construct local approximation spaces with exponentially decreasing error in the pre-asymtotic regime.

Multiscale convergence and reiterated homogenization of parabolic problems

Applications of Mathematics, 2005

Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://project.dml.cz 50 (2005) APPLICATIONS OF MATHEMATICS No. 2, 131-151

Numerical approximation of a parabolic problem with nonlinear boundary condition in several space dimensions

2015

In this paper we study the asymptotic behaviour of a semidiscrete numerical approximation for the heat equation, ut = ∆u, in a bounded smooth domain, with a nonlinear flux boundary condition at the boundary, ∂u ∂η = up. We focus in the behaviour of blowing up solutions. First we prove that every numerical solution blows up in finite time if and only if p> 1 and that the numerical blow-up time converges to the continuous one as the mesh parameter goes to zero. Next, we show that the blow-up rate for the numerical scheme is different from the continuous one. Nevertheless we find that the blow-up set for the numerical approximations it is contained in a neighborhood of the blow-up set of the continuous problem, when the mesh parameter is small enough.

Global Lipschitz regularizing effects for linear and nonlinear parabolic equations

Journal de Mathématiques Pures et Appliquées, 2013

In this paper we prove global bounds on the spatial gradient of viscosity solutions to second order linear and nonlinear parabolic Cauchy problems in (0, T) × R N. Our assumptions include the case that the coefficients be both unbounded and with very mild local regularity (possibly weaker than the Dini continuity), the estimates only depending on the parabolicity constant and on the modulus of continuity of coefficients (but not on their L ∞-norm). Our proof provides the analytic counterpart to the probabilistic proof used in Priola and Wang [PW06] (J. Funct. Anal. 2006) to get this type of gradient estimates in the linear case. We actually extend such estimates to the case of possibly unbounded data and solutions as well as to the case of nonlinear operators including Bellman-Isaacs equations. We investigate both the classical regularizing effect (at time t > 0) and the possible conservation of Lipschitz regularity from t = 0, and similarly we prove global Hölder estimates under weaker assumptions on the coefficients. The estimates we prove for unbounded data and solutions seem to be new even in the classical case of linear equations with bounded and Hölder continuous coefficients. Finally, we compare in an appendix the analytic and the probabilistic approach discussing the analogy between the doubling variables method of viscosity solutions and the probabilistic coupling method. Contents 42 4.5. Examples and applications 45 4.5a Bellman-Isaacs operators 45 4.5b The case of nonlinear Hamiltonians 46 4.5c A Liouville type theorem 48 4.5d A remark on possible existence results 50 Appendix A. Probabilistic Vs analytic approach 50 References 54 This work has been supported by the Italian GNAMPA project 2008 "Problemi di Diffusione degeneri" and GNAMPA project 2010 "Proprietà di regolarità in Equazioni alle Derivate Parziali nonlineari legate a problemi di controllo".

Numerical approximation of a parabolic problem with a nonlinear boundary condition in several space dimensions

2002

In this paper we study the asymptotic behavior of a semidiscrete numerical approximation for the heat equation, ut = ∆u, in a bounded smooth domain with a nonlinear flux boundary condition, ∂u ∂η = u p . We focus in the behavior of blowing up solutions. We prove that every numerical solution blows up in finite time if and only if p > 1 and that the numerical blow-up time converges to the continuous one as the mesh parameter goes to zero. Also we show that the blow-up rate for the numerical scheme is different from the continuous one. Nevertheless we find that the blow-up set for the numerical approximations is contained in a small neighborhood of the blow-up set of the continuous problem when the mesh parameter is small enough.

Approximation of parabolic problems on grids locally refined in time and space

Applied Numerical Mathematics, 1994

We present a strategy for solving time-dependent problems on grids with local re nement in time and in space, using di erent time-space sizes in di erent regions of space. We discuss two approximations based on the discontinuous Galerkin method and the nite di erence method with piecewise constant and piecewise linear interpolation in the time direction along the interface between the coarse-and negrid regions. Next, we present an iterative method for solving the composite-grid system that is based on domain decomposition techniques and reduces to solution of standard problems with standard time stepping (alternatively on the coarse and ne grids). Finally, numerical results that con rm both the analysis and the convergence theory of the iterative method are presented.