The Ritz method for solving partial differential equations using number-theoretic grids

Abstract

Consider the problem
𝐿𝑢(⃗𝑥) = 𝑓(⃗𝑥),
𝑢(⃗𝑥)⃒⃒𝜕𝐺𝑠= 𝑔(⃗𝑥),
where 𝑓(⃗𝑥), 𝑔(⃗𝑥) ∈ 𝐸𝛼
𝑠 , 𝐿 is a linear differential operator with constant coefficients, 𝐺𝑠 is the
unit cube [0; 1]𝑠.
Its solution is reduced to finding the minimum of the functional
𝑣(𝑢(⃗𝑥)) =∫︁. . .𝐺𝑠∫︁ 𝐹 (⃗𝑥, 𝑢, 𝑢𝑥1 , . . . , 𝑢𝑥𝑠 ) 𝑑𝑥1 𝑙𝑑𝑜𝑡𝑠𝑑𝑥𝑠
under given boundary conditions.
The values of the functional 𝑣(𝑢(⃗𝑥)) in the Ritz method are considered not on the set of all admissible functions 𝑢(⃗𝑥), but on linear combinations 𝑢(⃗𝑥) = 𝑊0(⃗𝑥) + Σ︁(𝑛 𝑘=1) 𝑤𝑘𝑊𝑘(⃗𝑥),
where 𝑊𝑘(⃗𝑥) are some basic functions that we will find using number-theoretic interpolation, and 𝑊0(⃗𝑥) is a function that satisfies the given boundary conditions, and the rest 𝑊𝑘(⃗𝑥) satisfy
homogeneous boundary conditions.
On these polynomials, this functional turns into a function 𝜙( ⃗𝑤) of the coefficients 𝑤1, . . . ,𝑤𝑛. These coefficients are chosen so that the function 𝜙( ⃗𝑤) reaches an extremum.
Under some restrictions on the functional 𝑣(𝑢(⃗𝑥)) and the basis functions 𝑊𝑘(⃗𝑥), we obtain an approximate solution of the boundary value problem.