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A new semi-discrete finite element scheme for the evolution of threeparametrized curves by curvature flow that are connected by a triple junctionis presented and analyzed. In this triple junction, conditions are imposed onthe angles at which the curves meet. One of the key problems in analyzingmotion of networks by curvature law is the choice of a tangential velocity thatallows for motion of the triple junction, does not lead to mesh degeneration,and is amenable to an error analysis. Our approach consists in considering aperturbation of a classical smooth formulation. The problem we propose admits anatural variational formulation that can be discretized with finite elements.The perturbation can be made arbitrarily small when a regularization parametershrinks to zero. Convergence of the new scheme including optimal errorestimates are proved. These results are supported by some numerical tests.Finally, the influence of the small regularization parameter on the propertiesof scheme and the accuracy of the results is numerically investigated.

A converging finite element scheme for motion by curvature of a network with a triple junction