97 research outputs found
Optimal Control of Convective FitzHugh-Nagumo Equation
We investigate smooth and sparse optimal control problems for convective
FitzHugh-Nagumo equation with travelling wave solutions in moving excitable
media. The cost function includes distributed space-time and terminal
observations or targets. The state and adjoint equations are discretized in
space by symmetric interior point Galerkin (SIPG) method and by backward Euler
method in time. Several numerical results are presented for the control of the
travelling waves. We also show numerically the validity of the second order
optimality conditions for the local solutions of the sparse optimal control
problem for vanishing Tikhonov regularization parameter. Further, we estimate
the distance between the discrete control and associated local optima
numerically by the help of the perturbation method and the smallest eigenvalue
of the reduced Hessian
An -Adaptive Newton-Galerkin Finite Element Procedure for Semilinear Boundary Value Problems
In this paper we develop an -adaptive procedure for the numerical
solution of general, semilinear elliptic boundary value problems in 1d, with
possible singular perturbations. Our approach combines both a prediction-type
adaptive Newton method and an -version adaptive finite element
discretization (based on a robust a posteriori residual analysis), thereby
leading to a fully -adaptive Newton-Galerkin scheme. Numerical experiments
underline the robustness and reliability of the proposed approach for various
examples.Comment: arXiv admin note: text overlap with arXiv:1408.522
A NSFD discretization of two-dimensional singularly perturbed semilinear convection-diffusion problems
Despite the availability of an abundant literature on singularly perturbed problems,
interest toward non-linear problems has been limited. In particular, parameter-uniform
methods for singularly perturbed semilinear problems are quasi-non-existent. In this
article, we study a two-dimensional semilinear singularly perturbed convection-diffusion
problems. Our approach requires linearization of the continuous semilinear problem
using the quasilinearization technique. We then discretize the resulting linear problems
in the framework of non-standard finite difference methods. A rigorous convergence
analysis is conducted showing that the proposed method is first-order parameter-uniform
convergent. Finally, two test examples are used to validate the theoretical findings
Pricing renewable energy certificates
Mestrado Bolonha em Mathematical FinanceThis thesis is concerned with a pricing model of renewable energy certificates. We study a stochastic model based on a system of forward-backward stochastic differential equations with two stochastic factors. The forward processes are the accumulated renewable energy certificates and the renewable energy production rate, while the back ward process is the certificate's price. This setting allows us to derive a non-linear partial differential equation for the price. The first step in finding a numerical solution for that equation is its linearization via a duality algorithm. To solve the obtained linearized problem, we then use a characteristics scheme in time with infinite differences discretization. Finally, the convergence of the algorithm is checked, alongside its ap- plication to a real world problem and an analysis of the equation's sensitivity to its parameters.info:eu-repo/semantics/publishedVersio
Edge-based nonlinear diffusion for finite element approximations of convection–diffusion equations and its relation to algebraic flux-correction schemes
The final publication is available at Springer via http://dx.doi.org/10.1007/s00211-016-0808-zFor the case of approximation of convection–diffusion equations using piecewise affine continuous finite elements a new edge-based nonlinear diffusion operator is proposed that makes the scheme satisfy a discrete maximum principle. The diffusion operator is shown to be Lipschitz continuous and linearity preserving. Using these properties we provide a full stability and error analysis, which, in the diffusion dominated regime, shows existence, uniqueness and optimal convergence. Then the algebraic flux correction method is recalled and we show that the present method can be interpreted as an algebraic flux correction method for a particular definition of the flux limiters. The performance of the method is illustrated on some numerical test cases in two space dimensions
Nonlinear Preconditioning: How to use a Nonlinear Schwarz Method to Precondition Newton's Method
For linear problems, domain decomposition methods can be used directly as
iterative solvers, but also as preconditioners for Krylov methods. In practice,
Krylov acceleration is almost always used, since the Krylov method finds a much
better residual polynomial than the stationary iteration, and thus converges
much faster. We show in this paper that also for non-linear problems, domain
decomposition methods can either be used directly as iterative solvers, or one
can use them as preconditioners for Newton's method. For the concrete case of
the parallel Schwarz method, we show that we obtain a preconditioner we call
RASPEN (Restricted Additive Schwarz Preconditioned Exact Newton) which is
similar to ASPIN (Additive Schwarz Preconditioned Inexact Newton), but with all
components directly defined by the iterative method. This has the advantage
that RASPEN already converges when used as an iterative solver, in contrast to
ASPIN, and we thus get a substantially better preconditioner for Newton's
method. The iterative construction also allows us to naturally define a coarse
correction using the multigrid full approximation scheme, which leads to a
convergent two level non-linear iterative domain decomposition method and a two
level RASPEN non-linear preconditioner. We illustrate our findings with
numerical results on the Forchheimer equation and a non-linear diffusion
problem
An hp-adaptive Newton-discontinuous-Galerkin finite element approach for semilinear elliptic boundary value problems
In this paper we develop an hp-adaptive procedure for the numerical solution of general second-order semilinear elliptic boundary value problems, with possible singular perturbation. Our approach combines both adaptive Newton schemes and an hp-version adaptive discontinuous Galerkin finite element discretisation, which, in turn, is based on a robust hp-version a posteriori residual analysis. Numerical experiments underline the robustness and reliability of the proposed approach for various examples
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