Elliptic systems with polynomial nonlinearity usually possess multiple solutions. In order to find multiple solutions, such elliptic systems are discretized by eigenfunction expansion method (EEM). Error analysis of the discretization is presented, which is different from the error analysis of EEM for scalar elliptic equations in three aspects: first, the choice of framework for the nonlinear operator and the corresponding isomorphism of the linearized operator; second, the definition of an auxiliary problem in deriving the relation between the L² norm and H¹ norm of the Ritz projection error; third, the bilinearity/nonbilinearity of the linearized variational forms. The symmetric homotopy for the discretized equations preserves not only D₄ symmetry, but also structural symmetry. With the symmetric homotopy, a filter strategy and a finite element Newton refinement, multiple solutions to a system of semilinear elliptic equations arising from Bose–Einstein condensate are found.

中文翻译：

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在具有多项式非线性的椭圆系统中找到多个解

具有多项式非线性的椭圆系统通常具有多个解。为了找到多个解，这种椭圆系统通过特征函数展开法（EEM）离散化。提出了离散化的误差分析，它与标量椭圆方程的EEM误差分析在三个方面不同：第一，非线性算子的框架选择和线性化算子的相应同构。其次，在导出Ritz投影误差的L ²范数和H ¹范数之间的关系时，定义一个辅助问题。第三，线性化变分形式的双线性/非双线性。离散方程的对称同伦不仅保留D₄对称，也有结构对称。通过对称同伦，滤波策略和牛顿有限元精细化处理，发现了由玻色-爱因斯坦凝聚物产生的半线性椭圆方程组的多重解。