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In this paper, the fully discrete orthogonal collocation method for Sobolev equations is considered, and the equivalence for discrete Garlerkin method is proved. Optimal order error estimate is obtained.

Sobolev equations are a class of mathematical physics equations, which are widely used in engineering field. Many numerical methods have been proposed, such as the characteristic difference method [^{1}-Galerkin Finite Element Method [

{ u t = ∇ ( a ∇ u t + b ∇ u ) + f ( x , y , t ) , ( x , y ) ∈ Ω , t ∈ ( 0 , T ] , u | ∂ Ω = 0 , ( x , y ) ∈ ∂ Ω , t ∈ [ 0 , T ] , u | t = 0 = u 0 ( x , y ) , ( x , y ) ∈ Ω . (1)

In the equations, u t is the time derivative of u, and ∇ u is the gradient of u. Ω = [ 0 , 1 ] × [ 0 , 1 ] , ∂ Ω is the border of Ω . a = a ( x , y , t ) and b = b ( x , y , t ) are known bounded differentiable functions.

First, time is divided into n equal parts. Let Δ t = T n be the time step. Then we

introduce the following notations:

t n = n Δ t , u n = u ( t n ) , ∂ t u n = u n − u n − 1 Δ t , ∇ u n = ∇ u ( t n ) , ∂ t ∇ u n = ∇ u n − ∇ u n − 1 Δ t .

Then we discrete the spatial region Ω into grids by points ( x i , y j ) , i = 0 , 1 , 2 , ⋯ , M , j = 0 , 1 , 2 , ⋯ , N and x i , y j are satisfied 0 = x 0 < x 1 < ⋯ < x M = 1 , 0 = y 0 < y 1 < ⋯ < y N = 1 . Let [

Ω i j = [ x i − 1 , x i ] × [ y j − 1 , y j ] , h x i = x i − x i − 1 , h y j = y j − y j − 1 , h = m a x { h x , h y } , H 3 = { v = v ( x , y ) ∈ C 1 ( Ω ) | v | Ω i j is a Bi-cubic Hermit polynomial } , H 3 0 = { v = v ( x , y ) ∈ H 3 | v | ∂ Ω = 0 } .

The four Gauss points ( x i k , y j l ) ; k , l = 1 , 2 in Ω i j are collocation points as follows: x i k = x i − 1 + h x i θ k , y j l = y j − 1 + h y j θ l , k , l = 1 , 2 , where θ 1 = ( 3 − 3 ) / 6 , θ 2 = ( 3 + 3 ) / 6 . Then the intermediate variable q = a ∇ u t + b ∇ u is introduced so that the orthogonal collocation scheme as follows can be established. Seeking ( U , Q ) : [ 0, T ] → H 3 × H 3 , such that

{ ∂ t U n − ∇ Q n − f n } ( x i k , y j l ) = 0 , { Q n − ( a n ∂ t ∇ U n + b n ∇ U n ) } ( x i k , y j l ) = 0 , (2)

{ U | ∂ Ω = 0 , t ∈ [ 0 , T ] , U | t = 0 = u 0 ( x , y ) .

Now we set the following notations [

〈 u , v 〉 = ∑ i = 1 M ∑ j = 1 N 〈 u , v 〉 i j = ∑ i = 1 M ∑ j = 1 N h x h y 4 ∑ k , l = 1 2 u v ( x i k , y j l ) , 〈 u , v 〉 x = ∑ i = 1 M 〈 u , v 〉 i x = ∑ i = 1 M h x 2 ∑ k = 1 2 u v ( x i k , y ) , 〈 u , v 〉 y = ∑ j = 1 N 〈 u , v 〉 j y = ∑ j = 1 N h y 2 ∑ l = 1 2 u v ( x , y j l ) , | ‖ u ‖ | i 2 = 〈 u , u 〉 i , | ‖ u ‖ | 2 = 〈 u , u 〉 . (3)

Next, we are going to prove existence and uniqueness of collocation solution and obtain the error estimate.

Consider the following discrete Galerkin scheme

{ 〈 ∂ t U n , z 1 〉 + 〈 ∇ Q n , ∇ z 1 〉 − 〈 f n , z 1 〉 = 0 , z 1 ∈ H 3 0 , 〈 Q n , z 2 〉 − 〈 a n ∂ t ∇ U n + b n ∇ U n , z 2 〉 = 0 , z 2 ∈ H 3 0 . (4)

Theorem 3.1: The solutions of (4) and (2) are equivalent, existent and unique.

Proof: From the Equation (3), it is clear that the solution of (2) must be the solution of (4).

Let { ζ l : l = 1 , 2 , ⋯ , 4 M N } = { ( x i k , y j l ) , i = 1 , ⋯ , M , j = 1 , ⋯ , N , k , l = 1 , 2 } , { Z i } 4 M N be a group base of H 3 0 . Thereupon ∀ U n ( x , y ) ∈ H 3 0 can be expressed as U n ( x , y ) = ∑ i = 1 4 M N β i n Z i ( x , y ) . So (2) and (4) can be written in the form as follows

F β n + G β n − 1 = R , C β n + D β n − 1 = S ,

F = ( F i j ) 4 M N × 4 M N , F i j = Z j ( ζ i ) − a n Δ Z j ( ζ i ) − b n Δ t Δ Z j ( ζ i ) ,

C = ( C i j ) 4 M N × 4 M N , C i j = 〈 Z j ( ζ i ) , Z i 〉 − a n 〈 Δ Z j ( ζ i ) , Z i 〉 − b n Δ t 〈 Δ Z j ( ζ i ) , Z i 〉 ,

where G , D are both matrixs of 4 M N × 4 M N and R , S are both vectors of 4 M N . Obviously the solution of equation F τ = 0 must be satisfied the equation C τ = 0 , when τ is a vectors of 4 M N . So F is nonsingular when C is nonsingular. Then the solutions of (2) and (4) are unique. To get the existence and uniqueness, we just need to prove A = ( A i j ) 4 M N × 4 M N where A i j = 〈 Z j ( ζ i ) − a Δ Z j ( ζ i ) , Z i 〉 is nonsingular when Δ t is sufficiently small. And the nonsingularity of A has been proved [

Next we will need to analyse the error estimate of (4).

Define interpolation operators ( P 1 , P 2 ) which satisfied the following conditions

W = P 1 u , v = W − U , η = u − W , V = P 2 q , w = V − Q , ξ = q − V ,

〈 ∇ ( q n − V n ) , z 〉 = 0 , ∀ z ∈ H 3 ,

〈 a n ∇ ( u t n − W t n ) + b n ∇ ( u n − W n ) , ∇ z 〉 = 0 , ∀ z ∈ H 3 ,

i.e., u − U = v + η , q − Q = w + ξ . Now we can get the error equations

{ 〈 η t n , z 1 〉 + 〈 ∂ t v n , z 1 〉 + 〈 r n , z 1 〉 + 〈 ξ n + w n , ∇ z 1 〉 = 0 , z 1 ∈ H 3 0 , 〈 ξ n + w n , z 2 〉 − 〈 a n ( ∇ r n + ∂ t ∇ v n ) , z 2 〉 − 〈 b n ∇ v n , z 2 〉 = 0 , z 2 ∈ H 3 0 . (5)

where r n = W t n − ∂ t W n , ∇ r n = ∇ W t n − ∂ t ∇ W n . Then there is the theorem as follows.

Theorem 4.1: If u(x,y) is the accurate solution of (1), U ( x , y ) is the solution of the orthogonal collocation method, and u ( x , y ) satisfies the condition [

‖ u n − U n ‖ ≤ O ( h 4 + Δ t ) , ‖ q n − Q n ‖ ≤ O ( h 4 + Δ t ) .

Proof: First, it is clearly for r n , ∇ r n that

| ‖ r n ‖ | 2 ≤ C Δ t ∫ t n − 1 t n | ‖ W t t ‖ | 2 d s , | ‖ ∇ r n ‖ | 2 ≤ C Δ t ∫ t n − 1 t n | ‖ ∇ W t t ‖ | 2 d s . (6)

Then let z 1 = v n , z 2 = ∇ v n in (5), the equations

{ 〈 η t n , v n 〉 + 〈 ∂ t v n , v n 〉 + 〈 r n , v n 〉 + 〈 ξ n + w n , ∇ v n 〉 = 0, 〈 ξ n + w n , ∇ v n 〉 − 〈 a n ( ∇ r n + ∂ t ∇ v n ) , ∇ v n 〉 − 〈 b n ∇ v n , ∇ v n 〉 = 0

can be got. It is easily calculated to see that

a n 〈 ∂ t ∇ v n , ∇ v n 〉 + 〈 ∂ t v n , v n 〉 = − a n 〈 ∇ r n , ∇ v n 〉 − b n 〈 ∇ v n , ∇ v n 〉 − 〈 r n , v n 〉 − 〈 η t n , v n 〉 .

Then through the Cauchy inequality, ε-inequality and 〈 ∂ t v n , v n 〉 ≥ 1 2 ∂ t | ‖ v n ‖ | 2 ,

and the functions a and b are bounded, it leads to the inequality

1 2 ∂ t | ‖ ∇ v n ‖ | 2 + 1 2 ∂ t | ‖ v n ‖ | 2 ≤ K | ‖ ∇ r n ‖ | 2 + ε | ‖ ∇ v n ‖ | 2 + | ‖ ∇ v n ‖ | 2 + K | ‖ r n ‖ | 2 + ε | ‖ v n ‖ | 2 + K | ‖ η t n ‖ | 2 + ε | ‖ v n ‖ | 2 .

The coefficients K , C both have nothing to do with h , Δ t in the upper equation and following proof. Add the inequality (6) and make summation to the series sum from n = 1 to n and multiply Δ t . Then

| ‖ ∇ v n ‖ | 2 + | ‖ v n ‖ | 2 ≤ K Δ t ∑ i = 0 n ( | ‖ η t i ‖ | 2 + | ‖ ∇ v i ‖ | 2 + | ‖ v i ‖ | 2 ) + K Δ t 2 ∫ 0 t n ( | ‖ ∇ W t t ‖ | 2 + | ‖ W t t ‖ | 2 ) d s

is obtained. So it follows from discrete Gronwall lemma that

| ‖ ∇ v n ‖ | 2 + | ‖ v n ‖ | 2 ≤ K Δ t ∑ i = 0 n ( | ‖ η t i ‖ | 2 ) + K Δ t 2 ∫ 0 t n ( | ‖ ∇ W t t ‖ | 2 + | | ‖ W t t ‖ | 2 ) d s (7)

if Δ t is small enough.

Second, let z 1 = ∂ t v n , z 2 = ∂ t ∇ v n in (5), the equations

{ 〈 η t n , ∂ t v n 〉 + 〈 ∂ t v n , ∂ t v n 〉 + 〈 r n , ∂ t v n 〉 + 〈 ξ n + w n , ∂ t ∇ v n 〉 = 0, 〈 ξ n + w n , ∂ t ∇ v n 〉 − 〈 a n ( ∇ r n + ∂ t ∇ v n ) , ∂ t ∇ v n 〉 − 〈 b n ∇ v n , ∂ t ∇ v n 〉 = 0,

can be got. It is easy to get

a n 〈 ∂ t ∇ v n , ∂ t ∇ v n 〉 + 〈 ∂ t v n , ∂ t v n 〉 = − a n 〈 ∇ r n , ∂ t ∇ v n 〉 − b n 〈 ∇ v n , ∂ t ∇ v n 〉 − 〈 r n , ∂ t v n 〉 − 〈 η t n , ∂ t v n 〉 .

Then through Cauchy inequality and ε-inequality, (6) and (7) it leads to the inequality

| ‖ ∂ t ∇ v n ‖ | 2 + | ‖ ∂ t v n ‖ | 2 ≤ K | ‖ η t n ‖ | 2 + K Δ t ∑ i = 1 n | ‖ η t i ‖ | 2 + K Δ t 2 ∫ 0 t n ( | ‖ ∇ W t t ‖ | 2 + | ‖ W t t ‖ | 2 ) d s , (8)

if Δ t is sufficiently small.

At last, let z 2 = w n in the second equation of (5), it can be expressed as 〈 ξ n + w n , w n 〉 − 〈 a n ( ∇ r n + ∂ t ∇ v n ) , w n 〉 − 〈 b n ∇ v n , w n 〉 = 0. (7) and (8) implies that

| ‖ w n ‖ | 2 ≤ K | ‖ ξ n ‖ | 2 + K | ‖ η t n ‖ | 2 + K Δ t ∑ i = 1 n | ‖ η t i ‖ | 2 + K Δ t 2 ∫ 0 t n ( | ‖ ∇ W t t ‖ | 2 + | ‖ W t t ‖ | 2 ) d s . (9)

The results

‖ η ‖ ≤ C h 4 ( ∑ i , j ‖ u ( 4 ) ‖ 2 ) 1 2 , ‖ η t ‖ ≤ C h 4 ( ∑ i , j ‖ u t ( 4 ) ‖ 2 ) 1 2 , ‖ ξ ‖ ≤ C h 4 ( ∑ i , j ‖ q ( 1 ) ‖ 2 ) 1 2 , ‖ ξ t ‖ ≤ C h 4 ( ∑ i , j ‖ q t ( 4 ) ‖ 2 ) 1 2 , (10)

can be obtained from lemma 1.6 in [

‖ u n − U n ‖ 2 ≤ ‖ η n ‖ 2 + K ‖ η t ‖ 2 + K Δ t 2 ∫ 0 t n ( | | | ∇ W t t | | | 2 + | | | W t t | | | 2 ) d s ≤ K 1 h 8 + K 1 Δ t 2 ,

‖ q n − Q n ‖ 2 ≤ K ‖ ξ n ‖ 2 + K ‖ η t n ‖ 2 + K ‖ η t ‖ 2 + K Δ t 2 ∫ 0 t n ( | | | ∇ W t t | | | 2 + | | | W t t | | | 2 ) d s ≤ K 2 h 8 + K 2 Δ t 2 ,

where K 1 and K 2 are constants which have nothing to do with h and η t . Thus the theorem is proved.

Sincere thanks to the Basic Subjects Fund and Science Foundations of China University of Petroleum (Beijing) (NO. 2462015YQ0604, NO. 2462015QZDX02).

Ma, N., Bian, W.L. and Lu, X.F. (2017) Fully Discrete Orthogonal Collocation Method of Sobolev Equations. Journal of Applied Mathematics and Physics, 5, 2354-2359. https://doi.org/10.4236/jamp.2017.512192