57 research outputs found
A Hybrid High-Order Method for a Class of Strongly Nonlinear Elliptic Boundary Value Problems
In this article, we design and analyze a Hybrid High-Order (HHO) finite
element approximation for a class of strongly nonlinear boundary value
problems. We consider an HHO discretization for a suitable linearized problem
and show its well-posedness using the Gardings type inequality. The essential
ingredients for the HHO approximation involve local reconstruction and
high-order stabilization. We establish the existence of a unique solution for
the HHO approximation using the Brouwer fixed point theorem and contraction
principle. We derive an optimal order a priori error estimate in the discrete
energy norm. Numerical experiments are performed to illustrate the convergence
histories.Comment: arXiv admin note: substantial text overlap with arXiv:2110.1557
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 hp-local discontinuous Galerkin method for some quasilinear elliptic boundary value problems of nonmonotone type
In this paper, an hp-local discontinuous Galerkin method is applied to a class of quasilinear elliptic boundary value problems which are of nonmonotone type. On hp-quasiuniform meshes, using the Brouwer fixed point theorem, it is shown that the discrete problem has a solution, and then using Lipschitz continuity of the discrete solution map, uniqueness is also proved. A priori error estimates in broken H1 norm and L2 norm which are optimal in h, suboptimal in p are derived. These results are exactly the same as in the case of linear elliptic boundary value problems. Numerical experiments are provided to illustrate the theoretical results
An hp-Local Discontinuous Galerkin method for Parabolic\ud Integro-Differential Equations
In this article, a priori error analysis is discussed for an hp-local discontinuous Galerkin (LDG) approximation to a parabolic integro-differential equation. It is shown that the L2 -norm of the gradient and the L2 -norm of the potential are optimal in the discretizing parameter h and suboptimal in the degree of polynomial p. Due to the presence of the integral term, an introduction of an expanded mixed type Ritz-Volterra projection helps to achieve optimal estimates. Further, it is observed that a negative norm estimate of the gradient plays a crucial role in our convergence analysis. As in the elliptic case, similar results on order of convergence are established for the semidiscrete method after suitably modifying the numerical fluxes. The optimality of these theoretical results is tested in a series of numerical experiments on two dimensional domains
An adaptive -discontinuous Galerkin approach for nonlinear convection-diffusion problems
summary:We deal with a numerical solution of nonlinear convection-diffusion equations with the aid of the discontinuous Galerkin method (DGM). We propose a new -adaptation technique, which is based on a combination of a residuum estimator and a regularity indicator. The residuum estimator as well as the regularity indicator are easily evaluated quantities without the necessity to solve any local problem and/or any reconstruction of the approximate solution. The performance of the proposed -DGM is demonstrated
Structures and waves in a nonlinear heat-conducting medium
The paper is an overview of the main contributions of a Bulgarian team of
researchers to the problem of finding the possible structures and waves in the
open nonlinear heat conducting medium, described by a reaction-diffusion
equation. Being posed and actively worked out by the Russian school of A. A.
Samarskii and S.P. Kurdyumov since the seventies of the last century, this
problem still contains open and challenging questions.Comment: 23 pages, 13 figures, the final publication will appear in Springer
Proceedings in Mathematics and Statistics, Numerical Methods for PDEs:
Theory, Algorithms and their Application
Numerical Methods for Hamilton-Jacobi-Bellman Equations
In this work we considered HJB equations, that arise from stochastic optimal control problems
with a finite time interval. If the diffusion is allowed to become degenerate, the solution cannot be
understood in the classical sense. Therefore one needs the notion of viscosity solutions. With some
stability and consistency assumptions, monotone methods provide the convergence to the viscosity
solution. In this thesis we looked at monotone finite difference methods, semi lagragian methods and
finite element methods for isotropic diffusion. In the last chapter we introduce the vanishing moment
method, a method not based on monotonicity
Numerical homogenization for nonlinear strongly monotone problems
In this work we introduce and analyze a new multiscale method for strongly
nonlinear monotone equations in the spirit of the Localized Orthogonal
Decomposition. A problem-adapted multiscale space is constructed by solving
linear local fine-scale problems which is then used in a generalized finite
element method. The linearity of the fine-scale problems allows their
localization and, moreover, makes the method very efficient to use. The new
method gives optimal a priori error estimates up to linearization errors. The
results neither require structural assumptions on the coefficient such as
periodicity or scale separation nor higher regularity of the solution. The
effect of different linearization strategies is discussed in theory and
practice. Several numerical examples including stationary Richards equation
confirm the theory and underline the applicability of the method
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