Error Constants for the Semi-Discrete Galerkin Approximation of the Linear Heat Equation

In this paper, we propose $$L^2(J;H^1_0(\Omega ))$$ L 2 ( J ; H 0 1 ( Ω ) ) and $$L^2(J;L^2(\Omega ))$$ L 2 ( J ; L 2 ( Ω ) ) norm error estimates that provide the explicit values of the error constants for the semi-discrete Galerkin approximation of the linear heat equation. The derivation of these error estimates shows the convergence of the approximation to the weak solution of the linear heat equation. Furthermore, explicit values of the error constants for these estimates play an important role in the computer-assisted existential proofs of solutions to semi-linear parabolic partial differential equations. In particular, the constants provided in this paper are better than the existing constants and, in a sense, the best possible.

On the Construction of Solutions to a Problem with a Free Boundary for the Non-linear Heat Equation

The construction of solutions to the problem with a free boundary for the non-linear heat equation which have the heat wave type is considered in the paper. The feature of such solutions is that the degeneration occurs on the front of the heat wave which separates the domain of positive values of the unknown function and the cold (zero) background. A numerical algorithm based on the boundary element method is proposed. Since it is difficult to prove the convergence of the algorithm due to the non-linearity of the problem and the presence of degeneracy the comparison with exact solutions is used to verify numerical results. The construction of exact solutions is reduced to integrating the Cauchy problem for ODE. A qualitative analysis of the exact solutions is carried out. Several computational experiments were performed to verify the proposed method

A non-conservation stochastic partial differential equation driven by anisotropic fractional Lévy random field

We study a non-conservation second-order stochastic partial differential equation (SPDE) driven by multi-parameter anisotropic fractional Lévy noise (AFLN) and under different initial and/or boundary conditions. It includes the time-dependent linear heat equation and quasi-linear heat equation under the fractional noise as special cases. Unique existence and expressions of solution to the equation are proved and constructed. An AFLN is defined as the derivative of an anisotropic fractional Lévy random field (AFLRF) in certain sense. Comparing with existing noise systems, our non-Gaussian fractional noises are essentially observed from random disturbances on system accelerations rather than from those on system moving velocities. In the process of proving our claims, there are three folds. First, we consider the AFLRF as the generalized functional of sample paths of a pure jump Lévy process. Second, we build Skorohod integration with respect to the AFLN by white noise approach. Third, by combining this noise analysis method with the conventional PDE solution techniques, we provide solid proofs for our claims.

A posteriori analysis of the spectral element discretization of a non linear heat equation

The paper deals with a posteriori analysis of the spectral element discretization of a non linear heat equation. The discretization is based on Euler’s backward scheme in time and spectral discretization in space. Residual error indicators related to the discretization in time and in space are defined. We prove that those indicators are upper and lower bounded by the error estimation.