The Fourier Spectral Method For Solving Two-dimensional Elliptic Partial Differential Equations With Dirichlet Boundary Condition

In this paper,we introduce the Fourier Spectral method for solving the second-order elliptic equations with the first boundary value problem.Through the Laplace equation gives the two-dimensional Fourier spectral method under the condition of the concrete application process.In this paper, consider the two-dimensional Elliptic equation whereλ> Ois a constant,f(x,y)is given as a proper smooth 2π-eriodic function. [2]gives the Fourier spectral method of one-dimensional periodic boundary problem and they get the ideal results.Assume uN is the approximate solution of the equation,for s:0≤s≤m,they have such results: where C is a constant and it has nothing to do with u, N.This paper studies two-dimensional and even high-dimensional cases, this method some new changes and the corresponding error estimates, and provides the corresponding calculation example.In this paper,C is a constant with nothing to do with u, N.For any positive number s,Sobolev space Hs(Ω) and its norm is as usual.AssumeΩ= (0,2π)n C (?)Rn,the points in Rn is presented by x= (x1,…,xn). As Q is a Cartesian space,so we can use as basis function,whereκ=(κ1,…,κn), and SN= Span{φκ(x)|-N≤κj≤N,j=1,…,n},we use PN:L2(Ω)→SN as orthogonal projection operator,in other words for any ,we have one and only one PNu∈SN,such that for ,we seek its generalized derivative and get whereα=(α1,…,αn),αi≥0,αis called multiple indicators and |α|= By the Department of orthogonal trigono-metric functions, we get Assume where is presented the sum of every component ofκfrom -N to N,we care for the approximation of uN to u. Lemma 1 we have as u∈Hm(Ω) and s:≤s≤m,C is a constant and it has nothing to do with u, N. Consider the two-dimensional Laplace equation with the first boundary value problem: where f(x, y) is given as a proper smooth 2π-periodic function. In the space of , we use the complex conjugate function of v multiply the equation (1.3) and integral it on theΩ, so we can get follows with first boundary value conditions: Assume so the variational of the problem (1.3), (1.4) is as follows:explore u∈U in order to haveLet's consider the convergence of the result to the true solution.By Lemma 1 we haveTheorem 1 where u∈Hpσ(σ≥1). As can be seen from the above equation,the smoother u(x, y) is,the faster uN(x, y) convergence.Especially,if u(x, y) is a infinitely dif-ferentiable periodic function,uN(x,y) convergence to u(x,y) faster than any finite power of So, also said that spectrum method has "exponential conver-gence".moreover,Fourier spectral method requires a periodic solution,Pairs of non-periodic cases, can be used for a cyclical expansion, However, will appear at the border point of interruption will result in undue turbulence.