Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  develop a stability and convergence theory for a Discontinuous Galerkin formulation (DG) of a highly indefinite Helmholtz problem in $$\mathbb R ^{d}$$ R d , $$d\in \{1,2,3\}$$ d ? { 1 , 2 , 3 } . The theory covers conforming as ll as non-conforming generalized finite element methods. In contrast to conventional Galerkin methods where a minimal resolution condition is necessary to guarantee the unique solvability, it is proved that the DG-method admits a unique solution under much aker conditions. As an application  present the error analysis for the $$hp$$ hp -version of the finite element method explicitly in terms of the mesh width $$h$$ h , polynomial degree $$p$$ p and wavenumber $$k$$ k . It is shown that the optimal convergence order estimate is obtained under the conditions that $$kh/\sqrt{p}$$ kh / p is sufficiently small and the polynomial degree $$p$$ p is at least $$O(\log k)$$ O ( log k ) . On regular meshes, the first condition is improved to the requirement that $$kh/p$$ kh / p be sufficiently small.
['Jens Markus Melenk', 'Asieh Parsania', 'Stefan A. Sauter']
I am writing a research paper on a idea General DG-Methods for Highly Indefinite Helmholtz Problems
[{"from": "human", "value": "I am writing a research paper on a idea General DG-Methods for Highly Indefinite Helmholtz Problems"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  develop a stability and convergence theory for a Discontinuous Galerkin formulation (DG) of a highly indefinite Helmholtz problem in $$\\mathbb R ^{d}$$ R d , $$d\\in \\{1,2,3\\}$$ d ? { 1 , 2 , 3 } . The theory covers conforming as ll as non-conforming generalized finite element methods. In contrast to conventional Galerkin methods where a minimal resolution condition is necessary to guarantee the unique solvability, it is proved that the DG-method admits a unique solution under much aker conditions. As an application  present the error analysis for the $$hp$$ hp -version of the finite element method explicitly in terms of the mesh width $$h$$ h , polynomial degree $$p$$ p and wavenumber $$k$$ k . It is shown that the optimal convergence order estimate is obtained under the conditions that $$kh/\\sqrt{p}$$ kh / p is sufficiently small and the polynomial degree $$p$$ p is at least $$O(\\log k)$$ O ( log k ) . On regular meshes, the first condition is improved to the requirement that $$kh/p$$ kh / p be sufficiently small."}]