Finite Element Method for Elliptic Problems
Finite Element Method for Elliptic Problems
A Particle-Partition of Unity Method for the Solution of Elliptic, Parabolic, and Hyperbolic PDEs
SIAM Journal on Scientific Computing
A Particle-Partition of Unity Method--Part II: Efficient Cover Construction and Reliable Integration
SIAM Journal on Scientific Computing
Meshfree Particle Methods in the Framework of Boundary Element Methods for the Helmholtz Equation
Journal of Scientific Computing
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We consider a mixed-boundary-value/interface problem for the elliptic operator P=-@?"i"j@?"i(a"i"j@?"ju)=f on a polygonal domain @W@?R^2 with straight sides. We endowed the boundary of @W partially with Dirichlet boundary conditions u=0 on @?"D@W, and partially with Neumann boundary conditions @?"i"j@n"ia"i"j@?"ju=0 on @?"N@W. The coefficients a"i"j are piecewise smooth with jump discontinuities across the interface @C, which is allowed to have singularities and cross the boundary of @W. In particular, we consider ''triple-junctions'' and even ''multiple junctions''. Our main result is to construct a sequence of Generalized Finite Element spaces S"n that yield ''h^m-quasi-optimal rates of convergence'', m=1, for the Galerkin approximations u"n@?S"n of the solution u. More precisely, we prove that @?u-u"n@?@?Cdim(S"n)^-^m^/^2@?f@?"H"^"m"^"-"^"1"("@W"), where C depends on the data for the problem, but not on f, u, or n and dim(S"n)-~. Our construction is quite general and depends on a choice of a good sequence of approximation spaces S"n^' on a certain subdomain W that is at some distance to the vertices. In case the spaces S"n^' are Generalized Finite Element spaces, then the resulting spaces S"n are also Generalized Finite Element spaces.