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We propose a low-rank method for solving the Helmholtz equation. Our approach is based on the WaveHoltz method, which computes Helmholtz solutions by applying a time-domain filter to the solution of a related wave equation. The wave equation is discretized by high-order multiblock summation-by-parts finite differences. In two dimensions we seek to compress the solution in matrix form, and in three dimensions using tensor trains. To control rank growth we use step-truncation during time stepping and a low-rank Anderson acceleration for the WaveHoltz fixed point iteration. We have carried out extensive numerical experiments demonstrating the convergence and efficacy of the iterative scheme for free- and half-space problems in two and three dimensions with constant and piecewise constant wave speeds.

We develop a high order accurate numerical method for solving the elastic wave equation in second-order form. We hybridize the computationally efficient Cartesian grid formulation of finite differences with geometrically flexible discontinuous Galerkin methods on unstructured grids by a penalty based technique. At the interface between the two methods, we construct projection operators for the pointwise finite difference solutions and discontinuous Galerkin solutions based on piecewise polynomials. In addition, we optimize the projection operators for both accuracy and spectrum. We prove that the overall discretization conserves a discrete energy, and verify optimal convergence in numerical experiments.

Addressing the intricate challenges in plane elasticity, especially with non-vanishing traction and complex geometries, requires innovative methods. This paper offers a novel approach, drawing inspiration from the Neumann problem for the inhomogeneous Cauchy-Riemann equations. Our method applies to domains conformally equivalent to a unit disk or an annulus, focusing on deriving explicit solutions for the displacement field rather than the stress tensor, which distinguishes it from most traditional approaches. We explore solutions for specific classical cases to demonstrate its efficacy, such as a cardioid domain, a ring domain with a shifted hole, and a gear-like structure. This work enhances the toolkit for researchers and practitioners tackling isotropic planar elastostatic challenges with a unified and flexible approach.

För att underlätta övergången mellan gymnasie- och universitetsmatematik samt förbättra genomströmningen på den första matematikkursen för civilingenjörer vid Umeå universitet, har vi genomfört ett pedagogiskt projekt uppdelat i två steg. Vi inledde med att analysera resultaten från diagnostiska prov för nybörjarstudenter på civilingenjörsprogrammen, vilket avslöjade centrala kunskapsluckor. Med insikter från denna analys, skapade vi sedan självrättande quiz i lärplattformen Canvas för att effektivt täcka dessa luckor. Quizzen, fördelade över studenternas första matematikkurs, Endimensionell analys 1, har redan visat lovande resultat. Vi har observerat en märkbar ökning i genomströmningen på kursen, vilket tyder på att vår interaktiva inlärningsstrategi är framgångsrik. Trots detta står vi inför vissa utmaningar, inklusive att engagera studenterna att deltaga i de frivilliga quizarna och att finjustera själva frågorna. Vi ser fram emot att förbättra dessa aspekter i nästa fas av projektet, med målet att ytterligare stärka studenternas matematikkunskaper och deras framgång som ingenjörsstudenter.