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Register a new userSemi-Lagrangian methods for parabolic problems in divergence form
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Semi-Lagrangian methods have traditionally been developed in the framework of hyperbolic equations, but several extensions of the Semi-Lagrangian approach to diffusion and advection--diffusion problems have been proposed recently. These extensions are mostly based on probabilistic arguments and share the common feature of treating second-order operators in trace form, which makes them unsuitable for mass conservative models like the classical formulations of turbulent diffusion employed in computational fluid dynamics. We propose here some basic ideas for treating second-order operators in divergence form. A general framework for constructing consistent schemes in one space dimension is presented, and a specific case of nonconservative discretization is discussed in detail and analysed. Finally, an extension to (possibly nonlinear) problems in an arbitrary number of dimensions is proposed. Although the resulting discretization approach is only of first order in time, numerical results in a number of test cases highlight the advantages of these methods for applications to computational fluid dynamics and their superiority over to more standard low order time discretization approaches.
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Parameter identification problems for partial differential equations are an important subclass of inverse problems. The parameter-to-state map, which maps the parameter of interest to the respective solution of the PDE or state of the system, plays the central role in the (usually nonlinear) forward operator. Consequently, one is interested in well-definedness and further analytic properties such as continuity and differentiability of this operator w.r.t. the parameter in order to make sure that techniques from inverse problems theory may be successfully applied to solve the inverse problem. In this work, we present a general functional analytic framework suited for the study of a huge class of parameter identification problems including a variety of elliptic boundary value problems (in divergence form) with Dirichlet, Neumann, Robin or mixed boundary conditions. In particular, we show that the corresponding parameter-to-state operators fulfil, under suitable conditions, the tangential cone condition, which is often postulated for numerical solution techniques. This framework particularly covers the inverse medium problem and an inverse problem that arises in terahertz tomography.
This paper is concerned with the recovery of (approximate) solutions to parabolic problems from incomplete and possibly inconsistent observational data, given on a time-space cylinder that is a strict subset of the computational domain under consideration. Unlike previous approaches to this and related problems our starting point is a regularized least squares formulation in a continuous infinite-dimensional setting that is based on stable variational time-space formulations of the parabolic PDE. This allows us to derive a priori as well as a posteriori error bounds for the recovered states with respect to a certain reference solution. In these bounds the regularization parameter is disentangled from the underlying discretization. An important ingredient for the derivation of a posteriori bounds is the construction of suitable Fortin operators which allow us to control oscillation errors stemming from the discretization of dual norms. Moreover, the variational framework allows us to contrive preconditioners for the discrete problems whose application can be performed in linear time, and for which the condition numbers of the preconditioned systems are uniformly proportional to that of the regularized continuous problem. In particular, we provide suitable stopping criteria for the iterative solvers based on the a posteriori error bounds. The presented numerical experiments quantify the theoretical findings and demonstrate the performance of the numerical scheme in relation with the underlying discretization and regularization.
In this paper, we present a numerical verification method of solutions for nonlinear parabolic initial boundary value problems. Decomposing the problem into a nonlinear part and an initial value part, we apply Nakaos projection method, which is based on the full-discrete finite element method with constructive error estimates, to the nonlinear part and use the theoretical analysis for the heat equation to the initial value part, respectively. We show some verified examples for solutions of nonlinear problems from initial value to the neighborhood of the stationary solutions, which confirm us the actual effectiveness of our method.
In this paper, we develop a computational multiscale to solve the parabolic wave approximation with heterogeneous and variable media. Parabolic wave approximation is a technique to approximate the full wave equation. One benefit of the method is that: one wave propagation direction can be taken as an evolution direction, and we then can discretize it using a classical scheme like Backward Euler. Consequently, we obtain a set of quasi-gas-dynamic (QGD) models with different heterogeneous permeability fields. Then, we employ constraint energy minimization generalized multiscale finite element method (CEM-GMsFEM) to perform spatial discretization for the problem. The resulting system can be solved by combining the central difference in time evolution. Due to the variable media, we apply the technique of proper orthogonal decomposition (POD) to further the dimension of the problem and solve the corresponding model problem in the POD space instead of in the whole multiscale space spanned by all possible multiscale basis functions. We prove the stability of the full discretization scheme and give the convergence analysis of the proposed approximation scheme. Numerical results verify the effectiveness of the proposed method.
We derive a posteriori error estimates in the $L_infty((0,T];L_infty(Omega))$ norm for approximations of solutions to linear para bolic equations. Using the elliptic reconstruction technique introduced by Makridakis and Nochetto and heat kernel estimates for linear parabolic pr oblems, we first prove a posteriori bounds in the maximum norm for semidiscrete finite element approximations. We then establish a posteriori bounds for a fully discrete backward Euler finite element approximation. The elliptic reconstruction technique greatly simplifies our development by allow ing the straightforward combination of heat kernel estimates with existing elliptic maximum norm error estimators.
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