کد مقاله کد نشریه سال انتشار مقاله انگلیسی نسخه تمام متن
4641783 1341319 2009 19 صفحه PDF دانلود رایگان
عنوان انگلیسی مقاله ISI
Analysis of a modified Schrödinger operator in 2D: Regularity, index, and FEM
موضوعات مرتبط
مهندسی و علوم پایه ریاضیات ریاضیات کاربردی
پیش نمایش صفحه اول مقاله
Analysis of a modified Schrödinger operator in 2D: Regularity, index, and FEM
چکیده انگلیسی

Let r=(x12+x22)1/2 be the distance function to the origin O∈R2O∈R2, and let us fix δ>0δ>0. We consider the “Schrödinger-type mixed boundary value problem” −Δu+δr−2u=f∈Hm−1(Ω)−Δu+δr−2u=f∈Hm−1(Ω) on a bounded polygonal domain Ω⊂R2Ω⊂R2. The singularity in the potential δr−2δr−2 severely limits the regularity of the solution uu. This affects the rate of convergence to uu of the finite element approximations uS∈SuS∈S obtained using a quasi-uniform sequence of meshes. We show that a suitable graded sequence of meshes recovers the quasi-optimal convergence rate ‖u−un‖H1(Ω)≤Cdim(Sn)−m/2‖f‖Hm−1(Ω)‖u−un‖H1(Ω)≤Cdim(Sn)−m/2‖f‖Hm−1(Ω), where SnSn are the FE spaces of continuous, piecewise polynomial functions of degree m≥1m≥1 associated to our sequence of meshes and un=uSn∈Snun=uSn∈Sn are the FE approximate solutions. This is in spite of the fact that u⁄∈Hm+1(Ω)u⁄∈Hm+1(Ω) in general. One of the main results of our paper is to show that the singularities due to the potential and the singularities due to the singularities of the domain or to the change in boundary conditions can be treated in the same way. Our proof is based on regularity and well-posedness results in weighted Sobolev spaces, with the weight taking into account all singularities (including the ones coming from the potential). Our regularity results apply also to operators with weaker singularities, like the Schrödinger operator −Δu+δr−1−Δu+δr−1, for which we also obtain Fredholm conditions and a formula for the index. Our a priori estimates also extend to piecewise smooth domains (i.e., curvilinear polygonal domains).

ناشر
Database: Elsevier - ScienceDirect (ساینس دایرکت)
Journal: Journal of Computational and Applied Mathematics - Volume 224, Issue 1, 1 February 2009, Pages 320–338
نویسندگان
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