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I am trying to solve the shallow water equations using finite element method. Can anyone explain me how to treat nonlinear term in the Galerkin equation?

so for example in the equation for the velocity we will have the term [itex]u\nabla v[/itex]

where u and v are the velocity components. For the u and v we have their finite element representations like

[itex]u = \sum_{i_1,N}u_iB_i,\ \ \ v=\sum_{i=1,N}v_iB_i[/itex]

hence [itex]\nabla v=\sum_{i=1,N}v_i\nabla B_i[/itex]

hence after getting the Galerkin equation (mutiplying by test function and integrating) we will have

[itex]\int u\nabla vB_kdx=\int \sum _{i=1,N} u_iB_i\sum _{i=1,N} v_i\nabla B_i B_kdx=\sum_{i,j=1,N}u_iv_j\int B_i\nabla B_jB_k dx[/itex]

is there any not to treat the nonliear term explicitly - by not having a double sum?