Galerkin Approximation of a Non-Linear Parabolic Interface Problem on Finite and Spectral Elements

Abstract

Nonlinear parabolic interface problems are frequently encountered in the modelling of physical processes which involved two or more materials with different properties. Research had focused largely on solving linear parabolic interface problems with the use of Finite Element Method (FEM). However, Spectral Element Method (SEM) for approximating nonlinear parabolic interface problems is scarce in literature. This work was therefore designed to give a theoretical framework for the convergence rates of finite and spectral element solutions of a nonlinear parabolic interface problem under certain regularity assumptions on the input data.

A nonlinear parabolic interface problem of the form

ut − ∇ · (a(x, u)∇u) = f(x, u) in Ω × (0, T]

with initial and boundary conditions

u(x, 0) = u0(x) , u(x, t) = 0 on ∂Ω × [0, T]

and interface conditions

[u]Γ = 0,

a(x, u) ∂u ∂n Γ = g(x, t)

was considered on a convex polygonal domain Ω ∈ R 2 with the assumption that the unknown function u(x, t) is of low regularity across the interface, where f : Ω × R → R, a : Ω × R → R are given functions and g : [0, T] → H2 (Γ) ∩ H1/2 (Γ) is the interface function. Galerkin weak formulation was used and the solution domain was discretised into quasi-uniform triangular elements after which the unknown function was approximated by piecewise linear functions on the finite elements. The time discretisation was based on Backward Difference Schemes (BDS). The implementation of this was based on predictor-corrector method due to the presence of nonlinear terms. A four-step linearised FEM-BDS was proposed and analysed to ease the computational stress and improve on the accuracy of the time-discretisation. On spectral elements, the formulation was based on Legendre polynomials evaluated at Gauss-Lobatto-Legendre points. The integrals involved were evaluated by numerical quadrature. The linear theories of interface and noninterface problems as well as Sobolev imbedding inequalities were used to obtain the a priori and the error estimates. Other tools used to obtain the error estimates were approximation properties of linear interpolation operators and projection operators. The a priori estimates of the weak solution were obtained with low regularity assumption on the solution across the interface, and almost optimal convergence rates of O

h

1 + 1 | log h| 1/2

and O

h 2

1 + 1 | log h| in the L 2 (0, T; H1 (Ω)) and L 2 (0, T; L 2 (Ω)) norms respectively were established for the spatially discrete scheme. Almost optimal convergence rates of O

k + h

1 + 1 | log h| and O

k + h 2

1 + 1 | log h| in the L 2 (0, T; H1 (Ω)) and L 2 (0, T; L 2 (Ω)) norms were obtained for the fully discrete scheme based on the backward Euler scheme, respectively for small mesh size h and time step k. Similar error estimates were obtained for two-step implicit scheme and four-step linearised FEM-BDS. The solution by SEM was found to converge spectrally in the L 2 (0, T; L 2 (Ω))-norm as the degree of the Legendre polynomial increases. Convergence rates of almost optimal order in the L 2 (0, T; H1 (Ω)) and L 2 (0, T; L 2 (Ω)) norms for finite element approximation of a nonlinear parabolic interface problem were established when the integrals involved were evaluated by numerical quadrature.