A Godunov-Mixed Finite Element Method on Changing Meshes for the Nonlinear Sobolev Equations

and Applied Analysis 3 estimates in L2-norm under very general changes in the mesh in Section 4. In Section 5, we present results of numerical experiment, which confirm our theoretical results. Throughout the analysis, the symbol K will denote a generic constant, which is independent of mesh parameters Δt and h and not necessarily the same at different occurrences. 2. The Godunov-Mixed Method on Changing Meshes At first we give some notation and basic assumptions. The usual Sobolev spaces and norms are adopted on Ω. The inner product on L2 Ω is denoted by f, g ∫ Ω fgdx. Define the following spaces and norms: H div,Ω { f ( fx, fy ) ; fx, fy,∇ · f ∈ H Ω } , m ≥ 0, ‖f‖Hm div ∥ fx ∥ ∥ 2 m ∥ fy ∥ ∥ 2 m ‖∇ · f‖m, H div,Ω H0 div,Ω , W L2 Ω { φ ≡ constant on Ω} , V {v ∈ H div;Ω | v · n 0 on ∂Ω,n is the unit outward norm to ∂Ω}. 2.1 Let Δt > 0 n 1, 2, . . . ,N∗ denote different time steps, t ∑n k 1 Δt , T ∑N∗ n 1 Δt , Δt maxnΔt. Assume that the time steps Δt do not change too rapidly; that is, we assume there exist positive constants t∗ and t∗ which are independent of n and Δt such that t∗ ≤ Δt n Δtn−1 ≤ t∗. 2.2 For a given function g x, t , let g g x, t . Assume Ω 0, 1 × 0, 1 . At each time level t, we construct a quasiuniform rectangular partition K h {en i } of Ω: δ x : 0 x n 1/2 < x n 3/2 < · · · < x i 1/2 < · · · < x In 1/2 1, δ y : 0 y n 1/2 < y n 3/2 < · · · < y j 1/2 < · · · < y Jn 1/2 1. 2.3 Let hni,x x n i 1/2 − x i−1/2, hni,y y j 1/2 − y j−1/2, h maxi,j{hi,x, hi,y}, and h maxnh. And x i x n i 1/2 x n i−1/2 /2 is the midpoint of x n i−1/2, x n i 1/2 , h n i 1/2,x x n i 1 − x i , Δt O h . Let y j , h n j 1/2,y be defined analogously. Let M −1 δ n x { f : f |Bn i,x ∈ P k ( B i,x ) , i 1, . . . , I } , M 0 δ n x M k −1 δ n x ∩ C0 0, 1 , 2.4 where B i,x x n i−1/2, x n i 1/2 and P k B i,x is the set of all polynomials of degree less than or equal to k defined on B i,x. Similar definitions are given to M 0 −1 δ n y and M 1 0 δ n y . 4 Abstract and Applied Analysis Then the lowest order Raviart-Thomas spaces W h and V n h are given by W h M 0 −1 δ n x ⊗M0 −1 ( δ y ) , V n h V n h,x × V n h,y, V n h,x M 1 0 δ n x ⊗M0 −1 ( δ y ) , V n h,y M 0 −1 δ n x ⊗M1 0 ( δ y ) . 2.5 That is, the space W h is the space of functions which are constant on each element e i ∈ K h , and V n h is the space of vector valued functions whose components are continuous and linear on each element e i ∈ K h . The degrees of freedom of a function v ∈ V n h correspond to the values of v · γ at the midpoints of the sides of e i , where γ is the unit outward normal to ∂e i . It is easy to see that W h × V n h ⊂ W × V and divV n h W h . By introducing variables z −∇u, z bz azt, g cu c1u, c2u T g1, g2 , and c x, t, u ∂c1/∂u, ∂c2/∂u , we modify the first equation in 1.1 as ut ∇ · z ∇ · g uc u · z f u . 2.6 Here we are using the so-called “expanded” mixed finite element method, proposed by Arbogast et al. 20 , which gives a gradient approximation z as well as an approximation to the diffusion term z. The weak form of 2.6 is ut,w ∇ · z, w ∇ · g, w uc u · z, w ( f u , w ) , 2.7