Bounds on Amplification Factors, Element-Node and Node-Node Equivalence for Square Finite Elements in Kinematic Shallow Water Equations

The upper and lower bounds of amplification factors of lumped finite element schemes are compared with nodal (integer or half-integer multiple of) eigen-value solutions of consistent finite element scheme at element and node levels of error analysis. The closeness or proximity between bounds on solutions of amplification factors and eigen-solutions reveals that the two methods, consistent and lumped finite element schemes are equivalent. The element error solutions of lumped mass matrix assumption and consistent nodal solution denotes the element-node error equivalence and the nodal solutions of all of the finite element schemes denote the node-node error equivalence for square finite elements in kinematic wave shallow water equations. The comparison plots of lumped and consistent finite element schemes are presented in this paper for illustration.

Arbitrarily small nodal solutions for nonhomogeneous Robin problems

In the present paper, we consider a nonlinear Robin problem driven by a nonhomogeneous differential operator and with a reaction which is only locally defined. Using cut-off techniques and variational tools, we show that the problem has a sequence of nodal solutions converging to zero in C 1 ( Ω ‾ ).