A quadratic optimal control problem governed by parabolic equations with integral constraints is considered. A fully discrete finite element scheme is constructed for the optimal control problem, with finite elements for the spatial but the backward Euler method for the time discretisation. Some superconvergence results of the control, the state and the adjoint state are proved. Some numerical ...

We prove the doubling property of L-caloric measure corresponding to the second order parabolic equation in the whole space and in Lipschitz domains. For parabolic equations in the divergence form, a weaker form of the doubling property follows easily from a recent result, the backward Harnack inequality, and known estimates of Green’s function. Our method works for both the divergence and nond...

In this paper we study the numerical solution for an p−Laplacian type of evolution system Ht + ∇ × [|∇ × H|p−2∇ × H] = F(x, t), p > 2 in two space dimensions. For large p this system is an approximation of Bean’s critical-state model for type-II superconductors. By introducing suitable transformation, the system is equivalent to a nonlinear parabolic equation. For the nonlinear parabolic proble...

In this paper we study the nonconvex anisotropic mean curvature flow of a hypersurface. This corresponds to an anisotropic mean curvature flow where the anisotropy has a nonconvex Frank diagram. The geometric evolution law is therefore forward-backward parabolic in character, hence ill-posed in general. We study a particular regularization of this geometric evolution, obtained with a nonlinear ...

A discretization of an optimal control problem a stochastic parabolic equation driven by multiplicative noise is analyzed. The state discretized the continuous piecewise linear element method in space and backward Euler scheme time. convergence rate $$ O(\tau ^{1/2} + h^2) rigorously derived.

Abstract In recent years, tremendous progress has been made on numerical algorithms for solving partial differential equations (PDEs) in a very high dimension, using ideas from either nonlinear (multilevel) Monte Carlo or deep learning. They are potentially free of the curse dimensionality many different applications and have proven to be so case some methods parabolic PDEs. this paper, we revi...

Using two models that incorporate a nonlinear forward-backward heat equation, we demonstrate the existence of well-deened weak solutions containing shocks for diiusive problems. Occurrence of shocks is connected to multivalued inverse solutions and non-monotone potential functions. Unique viscous solutions are determined from perturbation theory by matching to a shock layer condition. Results o...

• A nonlinear coupled parabolic free boundary problem modeling plaque growth in the artery, using finite difference-collocation method, is solved. We have fixed domain front fixing method and model simplified by changing mixed condition to a Neumann one applying suitable change of variables achieve more comfortable results for numerical analysis. Applying difference second order backward formul...

The classical Feynman-Kac formula states the connection between linear parabolic partial differential equations (PDEs), like the heat equation, and expectation of stochastic processes driven by Brownian motion. It gives then a method for solving linear PDEs by Monte Carlo simulations of random processes. The extension to (fully)nonlinear PDEs led in the recent years to important developments in...

In the paper, we study the global existence of weak solution of the fully nonlinear parabolic problem (1.1)-(1.3) with nonlinear boundary conditions for the situation without strong absorption terms. Also, we consider the blow up of global solution of the problem (1.1)-(1.3) by using the convexity method.